L(s) = 1 | + 16·2-s + 192·4-s − 114·5-s + 280·7-s + 2.04e3·8-s − 1.82e3·10-s − 5.18e3·11-s − 6.66e3·13-s + 4.48e3·14-s + 2.04e4·16-s − 3.65e4·17-s + 6.47e3·19-s − 2.18e4·20-s − 8.29e4·22-s + 1.29e4·23-s − 7.70e4·25-s − 1.06e5·26-s + 5.37e4·28-s − 1.60e4·29-s − 1.60e5·31-s + 1.96e5·32-s − 5.84e5·34-s − 3.19e4·35-s − 2.86e5·37-s + 1.03e5·38-s − 2.33e5·40-s + 5.38e5·41-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.407·5-s + 0.308·7-s + 1.41·8-s − 0.576·10-s − 1.17·11-s − 0.841·13-s + 0.436·14-s + 5/4·16-s − 1.80·17-s + 0.216·19-s − 0.611·20-s − 1.66·22-s + 0.222·23-s − 0.986·25-s − 1.18·26-s + 0.462·28-s − 0.122·29-s − 0.965·31-s + 1.06·32-s − 2.54·34-s − 0.125·35-s − 0.931·37-s + 0.306·38-s − 0.576·40-s + 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 3 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 + 114 T + 18011 p T^{2} + 114 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 40 p T + 555582 T^{2} - 40 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5184 T + 27915142 T^{2} + 5184 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6662 T + 19857231 T^{2} + 6662 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 36510 T + 886979635 T^{2} + 36510 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6472 T + 1580438790 T^{2} - 6472 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12960 T + 6740530894 T^{2} - 12960 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 16074 T + 27847652863 T^{2} + 16074 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 160136 T + 60626118030 T^{2} + 160136 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 286958 T + 70225805703 T^{2} + 286958 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 538284 T + 238942946710 T^{2} - 538284 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1770896 T + 1319815433094 T^{2} + 1770896 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 279960 T + 675632927470 T^{2} + 279960 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2373852 T + 3297400780606 T^{2} + 2373852 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 690336 T + 36525190438 T^{2} + 690336 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3688886 T + 9632325380127 T^{2} + 3688886 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 983096 T + 6122082589350 T^{2} + 983096 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2776920 T + 17581276612798 T^{2} + 2776920 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9891094 T + 46451222820267 T^{2} - 9891094 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5078336 T + 43554395985246 T^{2} + 5078336 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8946312 T + 49317386349190 T^{2} + 8946312 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 682566 T + 64419777386971 T^{2} + 682566 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18464252 T + 245717615436102 T^{2} + 18464252 p^{7} T^{3} + p^{14} T^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.21786114777776024704574728506, −11.15678338769032502615010897382, −10.55683458825876423715258499831, −9.883737037649512303963254585403, −9.385254345672848198460823637256, −8.547140531171669970809502672415, −7.951185358733706421688701489229, −7.61281178247341286892844047122, −6.85907934346537322917624868207, −6.57231370481043889180146270733, −5.66508122964979424408628785209, −5.24355510012601452437072835163, −4.51350789209724869015020629368, −4.43343468309845216018557350170, −3.25377743169347156441397066632, −3.03135978975550477732341401944, −1.94561446053029755950027269177, −1.77485072920369584648414482825, 0, 0,
1.77485072920369584648414482825, 1.94561446053029755950027269177, 3.03135978975550477732341401944, 3.25377743169347156441397066632, 4.43343468309845216018557350170, 4.51350789209724869015020629368, 5.24355510012601452437072835163, 5.66508122964979424408628785209, 6.57231370481043889180146270733, 6.85907934346537322917624868207, 7.61281178247341286892844047122, 7.951185358733706421688701489229, 8.547140531171669970809502672415, 9.385254345672848198460823637256, 9.883737037649512303963254585403, 10.55683458825876423715258499831, 11.15678338769032502615010897382, 11.21786114777776024704574728506