| L(s) = 1 | − 2-s + 4-s + 5-s − 5·7-s − 8-s − 10-s − 4·11-s − 2·13-s + 5·14-s + 16-s + 2·17-s − 8·19-s + 20-s + 4·22-s + 3·23-s − 4·25-s + 2·26-s − 5·28-s − 4·31-s − 32-s − 2·34-s − 5·35-s + 4·37-s + 8·38-s − 40-s + 2·41-s + 6·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s − 0.554·13-s + 1.33·14-s + 1/4·16-s + 0.485·17-s − 1.83·19-s + 0.223·20-s + 0.852·22-s + 0.625·23-s − 4/5·25-s + 0.392·26-s − 0.944·28-s − 0.718·31-s − 0.176·32-s − 0.342·34-s − 0.845·35-s + 0.657·37-s + 1.29·38-s − 0.158·40-s + 0.312·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.70631100130705, −13.33498360555867, −12.87011947626038, −12.46399659370966, −12.32016035393010, −11.30840783958708, −10.85826653591804, −10.38014388286903, −10.08415311492241, −9.520962273311126, −9.239219616425030, −8.712193960863468, −8.025894949759027, −7.546585173952672, −7.088523026351899, −6.499916642930191, −5.999664825834760, −5.753275085606184, −4.984274233081280, −4.235196747241678, −3.615722354358223, −2.948469634701183, −2.488556194800431, −2.092179525860999, −1.041441309567272, 0, 0,
1.041441309567272, 2.092179525860999, 2.488556194800431, 2.948469634701183, 3.615722354358223, 4.235196747241678, 4.984274233081280, 5.753275085606184, 5.999664825834760, 6.499916642930191, 7.088523026351899, 7.546585173952672, 8.025894949759027, 8.712193960863468, 9.239219616425030, 9.520962273311126, 10.08415311492241, 10.38014388286903, 10.85826653591804, 11.30840783958708, 12.32016035393010, 12.46399659370966, 12.87011947626038, 13.33498360555867, 13.70631100130705
