| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 4·7-s − 8-s + 9-s + 10-s + 5·11-s − 12-s − 13-s + 4·14-s + 15-s + 16-s − 18-s − 19-s − 20-s + 4·21-s − 5·22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 4·28-s − 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.872·21-s − 1.06·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.755·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−13.93298727308061, −13.10148875007521, −12.85486463297099, −12.26339030526579, −11.91012040365581, −11.52823654688746, −10.84249229341932, −10.45238984708028, −9.963175081321436, −9.378704720317282, −8.966624301628066, −8.810287896783167, −7.730679747267586, −7.359664549409214, −6.891355909470464, −6.437314690394729, −5.939227408350214, −5.543694702549395, −4.534592224166945, −4.037285245844584, −3.517965239300739, −2.983621294221220, −2.135191572425004, −1.419322140795444, −0.6418688682599047, 0,
0.6418688682599047, 1.419322140795444, 2.135191572425004, 2.983621294221220, 3.517965239300739, 4.037285245844584, 4.534592224166945, 5.543694702549395, 5.939227408350214, 6.437314690394729, 6.891355909470464, 7.359664549409214, 7.730679747267586, 8.810287896783167, 8.966624301628066, 9.378704720317282, 9.963175081321436, 10.45238984708028, 10.84249229341932, 11.52823654688746, 11.91012040365581, 12.26339030526579, 12.85486463297099, 13.10148875007521, 13.93298727308061
