| L(s) = 1 | + 12·3-s − 18·5-s + 28·7-s + 51·9-s − 44·11-s − 134·13-s − 216·15-s − 74·17-s + 164·19-s + 336·21-s − 194·23-s − 69·25-s + 98·27-s − 108·29-s + 412·31-s − 528·33-s − 504·35-s + 286·37-s − 1.60e3·39-s − 18·41-s + 496·43-s − 918·45-s − 62·47-s + 490·49-s − 888·51-s − 828·53-s + 792·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.60·5-s + 1.51·7-s + 17/9·9-s − 1.20·11-s − 2.85·13-s − 3.71·15-s − 1.05·17-s + 1.98·19-s + 3.49·21-s − 1.75·23-s − 0.551·25-s + 0.698·27-s − 0.691·29-s + 2.38·31-s − 2.78·33-s − 2.43·35-s + 1.27·37-s − 6.60·39-s − 0.0685·41-s + 1.75·43-s − 3.04·45-s − 0.192·47-s + 10/7·49-s − 2.43·51-s − 2.14·53-s + 1.94·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.703532697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.703532697\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 4 p T + 31 p T^{2} - 602 T^{3} + 1168 p T^{4} - 602 p^{3} T^{5} + 31 p^{7} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 18 T + 393 T^{2} + 758 p T^{3} + 56148 T^{4} + 758 p^{4} T^{5} + 393 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 134 T + 10520 T^{2} + 559034 T^{3} + 26667070 T^{4} + 559034 p^{3} T^{5} + 10520 p^{6} T^{6} + 134 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 74 T + 17468 T^{2} + 831750 T^{3} + 118657126 T^{4} + 831750 p^{3} T^{5} + 17468 p^{6} T^{6} + 74 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 164 T + 11968 T^{2} - 138820 T^{3} - 15467058 T^{4} - 138820 p^{3} T^{5} + 11968 p^{6} T^{6} - 164 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 194 T + 46053 T^{2} + 5374478 T^{3} + 784861428 T^{4} + 5374478 p^{3} T^{5} + 46053 p^{6} T^{6} + 194 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 108 T + 42724 T^{2} - 1514908 T^{3} + 528463590 T^{4} - 1514908 p^{3} T^{5} + 42724 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 412 T + 51613 T^{2} + 11046366 T^{3} - 3907770320 T^{4} + 11046366 p^{3} T^{5} + 51613 p^{6} T^{6} - 412 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 286 T + 135513 T^{2} - 31935490 T^{3} + 9691467156 T^{4} - 31935490 p^{3} T^{5} + 135513 p^{6} T^{6} - 286 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 18 T + 114524 T^{2} - 18203666 T^{3} + 5875037446 T^{4} - 18203666 p^{3} T^{5} + 114524 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 496 T + 308268 T^{2} - 103155312 T^{3} + 35378135478 T^{4} - 103155312 p^{3} T^{5} + 308268 p^{6} T^{6} - 496 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 62 T + 345832 T^{2} + 21522086 T^{3} + 50715680878 T^{4} + 21522086 p^{3} T^{5} + 345832 p^{6} T^{6} + 62 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 828 T + 705892 T^{2} + 360136820 T^{3} + 165458948486 T^{4} + 360136820 p^{3} T^{5} + 705892 p^{6} T^{6} + 828 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1224 T + 1275413 T^{2} - 804281630 T^{3} + 441199752912 T^{4} - 804281630 p^{3} T^{5} + 1275413 p^{6} T^{6} - 1224 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 350 T + 430588 T^{2} + 138974618 T^{3} + 88699741846 T^{4} + 138974618 p^{3} T^{5} + 430588 p^{6} T^{6} + 350 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1498 T + 1714917 T^{2} - 1237137334 T^{3} + 793168291468 T^{4} - 1237137334 p^{3} T^{5} + 1714917 p^{6} T^{6} - 1498 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2326 T + 3445789 T^{2} + 3267789594 T^{3} + 2320231641060 T^{4} + 3267789594 p^{3} T^{5} + 3445789 p^{6} T^{6} + 2326 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1630 T + 2286292 T^{2} + 1913055042 T^{3} + 1441611662422 T^{4} + 1913055042 p^{3} T^{5} + 2286292 p^{6} T^{6} + 1630 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 1020 T + 1989936 T^{2} - 1295128300 T^{3} + 1427479601566 T^{4} - 1295128300 p^{3} T^{5} + 1989936 p^{6} T^{6} - 1020 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1920 T + 2662892 T^{2} - 2752584832 T^{3} + 2433754978902 T^{4} - 2752584832 p^{3} T^{5} + 2662892 p^{6} T^{6} - 1920 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 1550 T + 1862281 T^{2} - 874387910 T^{3} + 705085008356 T^{4} - 874387910 p^{3} T^{5} + 1862281 p^{6} T^{6} - 1550 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2202 T + 5018625 T^{2} + 6116849198 T^{3} + 7416588769820 T^{4} + 6116849198 p^{3} T^{5} + 5018625 p^{6} T^{6} + 2202 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.79049005049065676486821112400, −6.11325098309941533167303827453, −6.07633075650367756801034945215, −5.98634181039460040145962360302, −5.74798232953495470437783446747, −5.18772456039638925425717161194, −4.98071531500333786428085326663, −4.87111201137943504660084390387, −4.77289357626319325068913595923, −4.36862681366438678134819733004, −4.35433900103875526129779816371, −3.97296022465909289860470046441, −3.81530721510154809140877502706, −3.33365835942593853335432881803, −3.21678272456601636832384188498, −2.95668971419341524536170272093, −2.77555895884605066745416542991, −2.29716549056613927460769880254, −2.21277724232716814138879751755, −2.20755400427456565281873324598, −1.90720178597180734805306218556, −1.26810760069353515326893902981, −0.830574608514208413296298699731, −0.51053792493050557233320647549, −0.22214449369852328533142672289,
0.22214449369852328533142672289, 0.51053792493050557233320647549, 0.830574608514208413296298699731, 1.26810760069353515326893902981, 1.90720178597180734805306218556, 2.20755400427456565281873324598, 2.21277724232716814138879751755, 2.29716549056613927460769880254, 2.77555895884605066745416542991, 2.95668971419341524536170272093, 3.21678272456601636832384188498, 3.33365835942593853335432881803, 3.81530721510154809140877502706, 3.97296022465909289860470046441, 4.35433900103875526129779816371, 4.36862681366438678134819733004, 4.77289357626319325068913595923, 4.87111201137943504660084390387, 4.98071531500333786428085326663, 5.18772456039638925425717161194, 5.74798232953495470437783446747, 5.98634181039460040145962360302, 6.07633075650367756801034945215, 6.11325098309941533167303827453, 6.79049005049065676486821112400
