| L(s) = 1 | − 3·3-s − 4·5-s + 7-s + 6·9-s − 11-s − 13-s + 12·15-s + 6·19-s − 3·21-s + 7·23-s + 11·25-s − 9·27-s − 4·29-s − 7·31-s + 3·33-s − 4·35-s + 9·37-s + 3·39-s − 3·41-s − 4·43-s − 24·45-s − 7·47-s + 49-s + 4·55-s − 18·57-s + 10·59-s + 61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s + 0.377·7-s + 2·9-s − 0.301·11-s − 0.277·13-s + 3.09·15-s + 1.37·19-s − 0.654·21-s + 1.45·23-s + 11/5·25-s − 1.73·27-s − 0.742·29-s − 1.25·31-s + 0.522·33-s − 0.676·35-s + 1.47·37-s + 0.480·39-s − 0.468·41-s − 0.609·43-s − 3.57·45-s − 1.02·47-s + 1/7·49-s + 0.539·55-s − 2.38·57-s + 1.30·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−9.162315483614103273787951987036, −8.019717144694446458650941793028, −7.32725700724389409434800004032, −6.87500603561565348190797158795, −5.57083895378343704241467893841, −4.99578374211428834490250749588, −4.23747306929926209571755248157, −3.21368858070487213139767950208, −1.12784722592657566804209320755, 0,
1.12784722592657566804209320755, 3.21368858070487213139767950208, 4.23747306929926209571755248157, 4.99578374211428834490250749588, 5.57083895378343704241467893841, 6.87500603561565348190797158795, 7.32725700724389409434800004032, 8.019717144694446458650941793028, 9.162315483614103273787951987036
