| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s − 2·9-s + 2·11-s + 12-s + 13-s + 3·14-s + 16-s + 2·17-s + 2·18-s − 3·21-s − 2·22-s + 6·23-s − 24-s − 26-s − 5·27-s − 3·28-s − 10·29-s + 2·31-s − 32-s + 2·33-s − 2·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.654·21-s − 0.426·22-s + 1.25·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s − 0.566·28-s − 1.85·29-s + 0.359·31-s − 0.176·32-s + 0.348·33-s − 0.342·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 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.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−8.260012878510816106301560866433, −7.42909035683074069935733464418, −6.76157233530584412983565958305, −6.04001267884665215301124319814, −5.31077066507981647086355901390, −3.90323238651177505293534889375, −3.26391052616546132545980998174, −2.56162541282049340289334602689, −1.34919402396069597600299115767, 0,
1.34919402396069597600299115767, 2.56162541282049340289334602689, 3.26391052616546132545980998174, 3.90323238651177505293534889375, 5.31077066507981647086355901390, 6.04001267884665215301124319814, 6.76157233530584412983565958305, 7.42909035683074069935733464418, 8.260012878510816106301560866433
