| L(s) = 1 | − 2-s + 4-s − 3·5-s + 4·7-s − 8-s + 3·10-s + 13-s − 4·14-s + 16-s + 3·17-s + 4·19-s − 3·20-s + 4·25-s − 26-s + 4·28-s − 9·29-s − 4·31-s − 32-s − 3·34-s − 12·35-s − 37-s − 4·38-s + 3·40-s + 8·43-s + 12·47-s + 9·49-s − 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 1.51·7-s − 0.353·8-s + 0.948·10-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.727·17-s + 0.917·19-s − 0.670·20-s + 4/5·25-s − 0.196·26-s + 0.755·28-s − 1.67·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 2.02·35-s − 0.164·37-s − 0.648·38-s + 0.474·40-s + 1.21·43-s + 1.75·47-s + 9/7·49-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−12.76776003469473, −12.36427945331245, −11.89833976614332, −11.58255258811565, −11.11047840707945, −10.85341582685868, −10.43956801700785, −9.634236793370600, −9.217974205454596, −8.811192899874838, −8.194569070990255, −7.825910563054710, −7.585460787472596, −7.234603536726562, −6.614487190332512, −5.704845353787427, −5.435707514299204, −4.955448643165827, −4.143921762240543, −3.809792825026583, −3.409541222644509, −2.475727584196405, −2.041999109161879, −1.186772495882782, −0.8875205512435259, 0,
0.8875205512435259, 1.186772495882782, 2.041999109161879, 2.475727584196405, 3.409541222644509, 3.809792825026583, 4.143921762240543, 4.955448643165827, 5.435707514299204, 5.704845353787427, 6.614487190332512, 7.234603536726562, 7.585460787472596, 7.825910563054710, 8.194569070990255, 8.811192899874838, 9.217974205454596, 9.634236793370600, 10.43956801700785, 10.85341582685868, 11.11047840707945, 11.58255258811565, 11.89833976614332, 12.36427945331245, 12.76776003469473
