| L(s) = 1 | − 2·3-s − 2·5-s + 9-s − 4·11-s − 2·13-s + 4·15-s − 6·17-s + 8·23-s − 25-s + 4·27-s + 2·29-s − 10·31-s + 8·33-s + 2·37-s + 4·39-s + 2·41-s + 4·43-s − 2·45-s − 4·47-s + 12·51-s − 6·53-s + 8·55-s + 6·59-s − 8·61-s + 4·65-s + 8·67-s − 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 1.03·15-s − 1.45·17-s + 1.66·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s − 1.79·31-s + 1.39·33-s + 0.328·37-s + 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 0.583·47-s + 1.68·51-s − 0.824·53-s + 1.07·55-s + 0.781·59-s − 1.02·61-s + 0.496·65-s + 0.977·67-s − 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| 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
−13.02449690770617, −12.77885983366473, −12.50302962832984, −11.79863720548162, −11.29367979758115, −11.14762542046671, −10.79990496982243, −10.25839381883633, −9.706382777117673, −8.990748652833513, −8.773525207006032, −8.035320140240917, −7.632294285352600, −7.080677564744043, −6.828031978643363, −6.123535832474980, −5.622769868615775, −5.104327038162893, −4.772943886312895, −4.290040825902439, −3.640027831920922, −2.893812668704130, −2.535471997973967, −1.723743449457141, −0.8535117436557705, 0, 0,
0.8535117436557705, 1.723743449457141, 2.535471997973967, 2.893812668704130, 3.640027831920922, 4.290040825902439, 4.772943886312895, 5.104327038162893, 5.622769868615775, 6.123535832474980, 6.828031978643363, 7.080677564744043, 7.632294285352600, 8.035320140240917, 8.773525207006032, 8.990748652833513, 9.706382777117673, 10.25839381883633, 10.79990496982243, 11.14762542046671, 11.29367979758115, 11.79863720548162, 12.50302962832984, 12.77885983366473, 13.02449690770617
