| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s − 3·13-s + 16-s − 3·17-s + 6·19-s + 3·20-s + 3·23-s + 4·25-s − 3·26-s + 6·29-s − 4·31-s + 32-s − 3·34-s − 2·37-s + 6·38-s + 3·40-s + 3·41-s − 6·43-s + 3·46-s + 4·50-s − 3·52-s + 9·53-s + 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s − 0.832·13-s + 1/4·16-s − 0.727·17-s + 1.37·19-s + 0.670·20-s + 0.625·23-s + 4/5·25-s − 0.588·26-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s − 0.328·37-s + 0.973·38-s + 0.474·40-s + 0.468·41-s − 0.914·43-s + 0.442·46-s + 0.565·50-s − 0.416·52-s + 1.23·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.172474067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.172474067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.69229993529641, −12.22862585576090, −11.84871613108595, −11.25009423443258, −10.90729385746875, −10.26873052424377, −9.918289754486810, −9.491725201278911, −9.185125067444473, −8.340034567769602, −8.164749295192180, −7.163136011826352, −6.939491969774382, −6.687974751347097, −5.858894111996297, −5.456249168872626, −5.221137398143873, −4.687820849552205, −4.052610911195163, −3.440054399866002, −2.844968717557980, −2.345371585742374, −1.994702237450695, −1.216222347124583, −0.6257557289621744,
0.6257557289621744, 1.216222347124583, 1.994702237450695, 2.345371585742374, 2.844968717557980, 3.440054399866002, 4.052610911195163, 4.687820849552205, 5.221137398143873, 5.456249168872626, 5.858894111996297, 6.687974751347097, 6.939491969774382, 7.163136011826352, 8.164749295192180, 8.340034567769602, 9.185125067444473, 9.491725201278911, 9.918289754486810, 10.26873052424377, 10.90729385746875, 11.25009423443258, 11.84871613108595, 12.22862585576090, 12.69229993529641
