| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s − 2·9-s + 3·10-s − 12-s + 5·13-s − 14-s + 3·15-s + 16-s + 2·18-s + 2·19-s − 3·20-s − 21-s + 6·23-s + 24-s + 4·25-s − 5·26-s + 5·27-s + 28-s − 3·30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.471·18-s + 0.458·19-s − 0.670·20-s − 0.218·21-s + 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.980·26-s + 0.962·27-s + 0.188·28-s − 0.547·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45662 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45662 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−14.89683869847564, −14.65350124758751, −13.81952838080484, −13.37387876517365, −12.53220600790545, −12.13824342860240, −11.64338011082415, −11.10474122118980, −10.92116755880993, −10.54324245430210, −9.401267911075502, −9.142520845235441, −8.462457260278907, −7.937912055910589, −7.744284639478502, −6.856200302944493, −6.424929529951458, −5.781584991991566, −5.152752593264253, −4.471673573471541, −3.817589795424014, −3.152144948924490, −2.610824566770977, −1.345429887378555, −0.8739486686816412, 0,
0.8739486686816412, 1.345429887378555, 2.610824566770977, 3.152144948924490, 3.817589795424014, 4.471673573471541, 5.152752593264253, 5.781584991991566, 6.424929529951458, 6.856200302944493, 7.744284639478502, 7.937912055910589, 8.462457260278907, 9.142520845235441, 9.401267911075502, 10.54324245430210, 10.92116755880993, 11.10474122118980, 11.64338011082415, 12.13824342860240, 12.53220600790545, 13.37387876517365, 13.81952838080484, 14.65350124758751, 14.89683869847564
