| L(s) = 1 | + 2-s − 15·3-s − 6·4-s − 15·6-s + 35·7-s + 2·8-s + 135·9-s + 66·11-s + 90·12-s − 2·13-s + 35·14-s + 11·16-s + 108·17-s + 135·18-s + 174·19-s − 525·21-s + 66·22-s − 116·23-s − 30·24-s − 2·26-s − 945·27-s − 210·28-s + 370·29-s + 342·31-s − 109·32-s − 990·33-s + 108·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 2.88·3-s − 3/4·4-s − 1.02·6-s + 1.88·7-s + 0.0883·8-s + 5·9-s + 1.80·11-s + 2.16·12-s − 0.0426·13-s + 0.668·14-s + 0.171·16-s + 1.54·17-s + 1.76·18-s + 2.10·19-s − 5.45·21-s + 0.639·22-s − 1.05·23-s − 0.255·24-s − 0.0150·26-s − 6.73·27-s − 1.41·28-s + 2.36·29-s + 1.98·31-s − 0.602·32-s − 5.22·33-s + 0.544·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.629413642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.629413642\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{5} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 - T + 7 T^{2} - 15 T^{3} + 3 p^{4} T^{4} - p^{5} T^{5} + 3 p^{7} T^{6} - 15 p^{6} T^{7} + 7 p^{9} T^{8} - p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 6 p T + 4555 T^{2} - 210928 T^{3} + 10802758 T^{4} - 383496700 T^{5} + 10802758 p^{3} T^{6} - 210928 p^{6} T^{7} + 4555 p^{9} T^{8} - 6 p^{13} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 2 T + 6833 T^{2} + 33080 T^{3} + 25238338 T^{4} + 84665324 T^{5} + 25238338 p^{3} T^{6} + 33080 p^{6} T^{7} + 6833 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 108 T + 7625 T^{2} - 580624 T^{3} + 28716998 T^{4} - 723515208 T^{5} + 28716998 p^{3} T^{6} - 580624 p^{6} T^{7} + 7625 p^{9} T^{8} - 108 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 174 T + 30655 T^{2} - 3260120 T^{3} + 355132498 T^{4} - 29134734196 T^{5} + 355132498 p^{3} T^{6} - 3260120 p^{6} T^{7} + 30655 p^{9} T^{8} - 174 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 + 116 T + 41159 T^{2} + 2945168 T^{3} + 716339614 T^{4} + 37811551864 T^{5} + 716339614 p^{3} T^{6} + 2945168 p^{6} T^{7} + 41159 p^{9} T^{8} + 116 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 370 T + 160385 T^{2} - 37125560 T^{3} + 8745786530 T^{4} - 1370740158092 T^{5} + 8745786530 p^{3} T^{6} - 37125560 p^{6} T^{7} + 160385 p^{9} T^{8} - 370 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 342 T + 142051 T^{2} - 26927448 T^{3} + 6639779138 T^{4} - 944601593732 T^{5} + 6639779138 p^{3} T^{6} - 26927448 p^{6} T^{7} + 142051 p^{9} T^{8} - 342 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 408 T + 163665 T^{2} + 51926944 T^{3} + 15115947738 T^{4} + 3482080406928 T^{5} + 15115947738 p^{3} T^{6} + 51926944 p^{6} T^{7} + 163665 p^{9} T^{8} + 408 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 802 T + 392457 T^{2} - 146273472 T^{3} + 44792512462 T^{4} - 12012058617212 T^{5} + 44792512462 p^{3} T^{6} - 146273472 p^{6} T^{7} + 392457 p^{9} T^{8} - 802 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 584 T + 413407 T^{2} + 149329184 T^{3} + 61499850522 T^{4} + 16229990754224 T^{5} + 61499850522 p^{3} T^{6} + 149329184 p^{6} T^{7} + 413407 p^{9} T^{8} + 584 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 716 T + 507419 T^{2} - 209415632 T^{3} + 90503203546 T^{4} - 27982753228744 T^{5} + 90503203546 p^{3} T^{6} - 209415632 p^{6} T^{7} + 507419 p^{9} T^{8} - 716 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 98 T + 503305 T^{2} - 71121272 T^{3} + 122661090658 T^{4} - 16369960190572 T^{5} + 122661090658 p^{3} T^{6} - 71121272 p^{6} T^{7} + 503305 p^{9} T^{8} - 98 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 704 T + 427487 T^{2} - 197865856 T^{3} + 99740405354 T^{4} - 55609053370240 T^{5} + 99740405354 p^{3} T^{6} - 197865856 p^{6} T^{7} + 427487 p^{9} T^{8} - 704 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 650 T + 689345 T^{2} - 375251480 T^{3} + 259020342850 T^{4} - 102268282520348 T^{5} + 259020342850 p^{3} T^{6} - 375251480 p^{6} T^{7} + 689345 p^{9} T^{8} - 650 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 180 T + 1024695 T^{2} + 16666000 T^{3} + 437663187690 T^{4} + 44169474067848 T^{5} + 437663187690 p^{3} T^{6} + 16666000 p^{6} T^{7} + 1024695 p^{9} T^{8} - 180 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 1470 T + 1881615 T^{2} - 1727909680 T^{3} + 1338943951990 T^{4} - 860023408325220 T^{5} + 1338943951990 p^{3} T^{6} - 1727909680 p^{6} T^{7} + 1881615 p^{9} T^{8} - 1470 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 534 T + 1342021 T^{2} + 930732232 T^{3} + 817882087762 T^{4} + 560574711779204 T^{5} + 817882087762 p^{3} T^{6} + 930732232 p^{6} T^{7} + 1342021 p^{9} T^{8} + 534 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 820 T + 1736795 T^{2} + 1027136240 T^{3} + 1389341527610 T^{4} + 657402503393464 T^{5} + 1389341527610 p^{3} T^{6} + 1027136240 p^{6} T^{7} + 1736795 p^{9} T^{8} + 820 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 1520 T + 3526775 T^{2} - 3535501120 T^{3} + 4405187467130 T^{4} - 3049079895261856 T^{5} + 4405187467130 p^{3} T^{6} - 3535501120 p^{6} T^{7} + 3526775 p^{9} T^{8} - 1520 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 286 T + 2177465 T^{2} - 600135040 T^{3} + 2162009088238 T^{4} - 560754412921604 T^{5} + 2162009088238 p^{3} T^{6} - 600135040 p^{6} T^{7} + 2177465 p^{9} T^{8} - 286 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 278 T + 3429709 T^{2} - 1008794696 T^{3} + 5371131513042 T^{4} - 1398135665763908 T^{5} + 5371131513042 p^{3} T^{6} - 1008794696 p^{6} T^{7} + 3429709 p^{9} T^{8} - 278 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.13347214760503841257494128878, −6.03504191458922841385725447513, −5.61653867899764130991224927091, −5.56222250386834418741691403411, −5.47779618243498133099448394630, −5.01651621987786428166546109785, −5.00309680217051552675187591869, −4.86036183477226738858204231223, −4.78188530861677088519054503989, −4.56207152204041803999835313809, −4.00670514305580666937731514627, −3.94696473929894485030788072505, −3.94077080021115760463172068318, −3.79313639410754705619385576345, −3.33287295743581480318361947979, −2.85234497190941740337044615919, −2.72019472199818740806466589942, −2.22790402894743411418134052724, −1.84286148409124367116133162101, −1.63840519329943269372013610292, −1.32330888947937907680771901432, −0.953974473112225349203123265015, −0.806578334579863178029107003067, −0.72886461652620949289329986726, −0.53953557024677900940052760144,
0.53953557024677900940052760144, 0.72886461652620949289329986726, 0.806578334579863178029107003067, 0.953974473112225349203123265015, 1.32330888947937907680771901432, 1.63840519329943269372013610292, 1.84286148409124367116133162101, 2.22790402894743411418134052724, 2.72019472199818740806466589942, 2.85234497190941740337044615919, 3.33287295743581480318361947979, 3.79313639410754705619385576345, 3.94077080021115760463172068318, 3.94696473929894485030788072505, 4.00670514305580666937731514627, 4.56207152204041803999835313809, 4.78188530861677088519054503989, 4.86036183477226738858204231223, 5.00309680217051552675187591869, 5.01651621987786428166546109785, 5.47779618243498133099448394630, 5.56222250386834418741691403411, 5.61653867899764130991224927091, 6.03504191458922841385725447513, 6.13347214760503841257494128878
