| L(s) = 1 | − 256·5-s + 1.53e3·9-s − 4.86e3·13-s + 7.16e3·17-s + 1.22e4·25-s − 9.24e4·29-s + 1.60e5·37-s − 2.62e5·41-s − 3.93e5·45-s + 1.40e5·49-s − 3.56e5·53-s + 9.89e5·61-s + 1.24e6·65-s − 3.32e5·73-s + 1.04e6·81-s − 1.83e6·85-s − 2.43e6·89-s + 2.49e6·97-s − 4.56e6·101-s + 1.59e5·109-s − 9.62e5·113-s − 7.47e6·117-s + 3.55e6·121-s + 1.78e6·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.04·5-s + 2.10·9-s − 2.21·13-s + 1.45·17-s + 0.784·25-s − 3.78·29-s + 3.15·37-s − 3.80·41-s − 4.31·45-s + 1.19·49-s − 2.39·53-s + 4.35·61-s + 4.53·65-s − 0.855·73-s + 1.97·81-s − 2.98·85-s − 3.45·89-s + 2.73·97-s − 4.42·101-s + 0.123·109-s − 0.667·113-s − 4.66·117-s + 2.00·121-s + 0.911·125-s + 7.76·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3078391152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3078391152\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 512 p T^{2} + 145618 p^{2} T^{4} - 512 p^{13} T^{6} + p^{24} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 128 T + 738 p^{2} T^{2} + 128 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 20124 p T^{2} + 14926782662 T^{4} - 20124 p^{13} T^{6} + p^{24} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3550720 T^{2} + 6441149921442 T^{4} - 3550720 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2432 T + 11115378 T^{2} + 2432 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 3584 T + 50975298 T^{2} - 3584 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 173944320 T^{2} + 11974040019863522 T^{4} - 173944320 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 41622724 T^{2} + 4717959955420230 T^{4} - 41622724 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 46208 T + 1612586802 T^{2} + 46208 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 314494212 T^{2} + 336230053380482822 T^{4} - 314494212 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 80000 T + 5717270418 T^{2} - 80000 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 131100 T + 13520186918 T^{2} + 131100 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4424436224 T^{2} + 77181140034285016482 T^{4} + 4424436224 p^{12} T^{6} + p^{24} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 39092918148 T^{2} + \)\(61\!\cdots\!02\)\( T^{4} - 39092918148 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 178048 T + 34363817298 T^{2} + 178048 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 123900290560 T^{2} + \)\(69\!\cdots\!62\)\( T^{4} - 123900290560 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 494720 T + 2241986202 p T^{2} - 494720 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 287598767616 T^{2} + \)\(35\!\cdots\!02\)\( T^{4} - 287598767616 p^{12} T^{6} + p^{24} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 285063901116 T^{2} + \)\(53\!\cdots\!46\)\( T^{4} + 285063901116 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 166400 T - 17887685022 T^{2} + 166400 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 460763935876 T^{2} + \)\(14\!\cdots\!10\)\( T^{4} - 460763935876 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 993491412480 T^{2} + \)\(45\!\cdots\!22\)\( T^{4} - 993491412480 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1217024 T + 1363939221282 T^{2} + 1217024 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 1248768 T + 1407050228738 T^{2} - 1248768 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−6.95406173175502072880922817206, −6.78346898469945775179363293395, −6.67928395744228285843904058689, −6.07804684926496380794530488569, −5.77827108403576513813690554462, −5.62956479595754235301302299746, −5.18416714651297134426934316070, −5.14105142575140593319973310514, −4.94036217672108196668808027296, −4.48611171320446341845381871467, −4.14106793641927651900253255453, −4.04134164767895511714727920736, −3.92019997601516013729430753742, −3.80255428595056528566011563140, −3.26037990954674395325121613005, −3.05283483301925368856878717385, −2.82906975743362332172379491325, −2.27220926915364391879749125965, −1.93174269925498702280181495822, −1.88766003985278767492925197762, −1.42706971246661492724945016832, −1.13659720461117704065452831914, −0.76279756533434827425072980173, −0.32952326126868895814598947814, −0.10353963090444705241417068915,
0.10353963090444705241417068915, 0.32952326126868895814598947814, 0.76279756533434827425072980173, 1.13659720461117704065452831914, 1.42706971246661492724945016832, 1.88766003985278767492925197762, 1.93174269925498702280181495822, 2.27220926915364391879749125965, 2.82906975743362332172379491325, 3.05283483301925368856878717385, 3.26037990954674395325121613005, 3.80255428595056528566011563140, 3.92019997601516013729430753742, 4.04134164767895511714727920736, 4.14106793641927651900253255453, 4.48611171320446341845381871467, 4.94036217672108196668808027296, 5.14105142575140593319973310514, 5.18416714651297134426934316070, 5.62956479595754235301302299746, 5.77827108403576513813690554462, 6.07804684926496380794530488569, 6.67928395744228285843904058689, 6.78346898469945775179363293395, 6.95406173175502072880922817206
