| L(s) = 1 | + 5-s − 7-s − 3·9-s − 2·13-s + 2·17-s − 8·19-s − 8·23-s + 25-s + 6·29-s − 35-s − 2·37-s − 6·41-s − 8·43-s − 3·45-s + 8·47-s + 49-s − 2·53-s − 8·59-s − 2·61-s + 3·63-s − 2·65-s − 8·67-s + 10·73-s + 16·79-s + 9·81-s − 16·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s − 0.554·13-s + 0.485·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.169·35-s − 0.328·37-s − 0.937·41-s − 1.21·43-s − 0.447·45-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.04·59-s − 0.256·61-s + 0.377·63-s − 0.248·65-s − 0.977·67-s + 1.17·73-s + 1.80·79-s + 81-s − 1.75·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−9.460703330298231211154271373757, −8.542650169130189195385773854839, −7.980400271570046670292479988256, −6.67870857385096784562453741977, −6.12134969234702219800483361799, −5.21303535245100746897960861432, −4.12805979189587433567035685655, −2.95019693479394245361088012637, −1.98739105123917122704963516918, 0,
1.98739105123917122704963516918, 2.95019693479394245361088012637, 4.12805979189587433567035685655, 5.21303535245100746897960861432, 6.12134969234702219800483361799, 6.67870857385096784562453741977, 7.980400271570046670292479988256, 8.542650169130189195385773854839, 9.460703330298231211154271373757
