| L(s) = 1 | − 3-s − 3·5-s − 2·7-s − 2·9-s + 11-s − 4·13-s + 3·15-s + 6·17-s − 8·19-s + 2·21-s + 3·23-s + 4·25-s + 5·27-s − 5·31-s − 33-s + 6·35-s − 37-s + 4·39-s + 10·43-s + 6·45-s − 3·49-s − 6·51-s − 6·53-s − 3·55-s + 8·57-s − 3·59-s − 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.755·7-s − 2/3·9-s + 0.301·11-s − 1.10·13-s + 0.774·15-s + 1.45·17-s − 1.83·19-s + 0.436·21-s + 0.625·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s − 0.174·33-s + 1.01·35-s − 0.164·37-s + 0.640·39-s + 1.52·43-s + 0.894·45-s − 3/7·49-s − 0.840·51-s − 0.824·53-s − 0.404·55-s + 1.05·57-s − 0.390·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.25900555383232209371156445284, −11.33372769095583858463760631653, −10.42805632650209012086523138057, −9.146174464181464239622953038476, −7.992911069953808459961425963145, −6.99346836799162715894329128020, −5.78719635391273558621520357914, −4.40400371694164943841525727402, −3.10139317797780286714650046035, 0,
3.10139317797780286714650046035, 4.40400371694164943841525727402, 5.78719635391273558621520357914, 6.99346836799162715894329128020, 7.992911069953808459961425963145, 9.146174464181464239622953038476, 10.42805632650209012086523138057, 11.33372769095583858463760631653, 12.25900555383232209371156445284
