| L(s) = 1 | + 2-s − 2·3-s − 4-s − 5-s − 2·6-s + 4·7-s − 3·8-s + 9-s − 10-s + 2·12-s + 13-s + 4·14-s + 2·15-s − 16-s − 2·17-s + 18-s + 6·19-s + 20-s − 8·21-s − 6·23-s + 6·24-s + 25-s + 26-s + 4·27-s − 4·28-s − 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 0.277·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s − 1.74·21-s − 1.25·23-s + 1.22·24-s + 1/5·25-s + 0.196·26-s + 0.769·27-s − 0.755·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.53910588371317865144147686599, −6.58571666443678942577577849907, −5.81818597598767129069878776659, −5.18563670224598709052489428190, −4.98122220833204581590039892928, −4.04925548353637447042354153452, −3.52545848522143013958310489719, −2.20184318885196204588692531253, −1.08672735030409724871085244058, 0,
1.08672735030409724871085244058, 2.20184318885196204588692531253, 3.52545848522143013958310489719, 4.04925548353637447042354153452, 4.98122220833204581590039892928, 5.18563670224598709052489428190, 5.81818597598767129069878776659, 6.58571666443678942577577849907, 7.53910588371317865144147686599
