| L(s) = 1 | + 2-s + 4-s + 2·5-s + 4·7-s + 8-s + 2·10-s − 2·11-s + 4·14-s + 16-s + 6·17-s + 3·19-s + 2·20-s − 2·22-s − 4·23-s − 25-s + 4·28-s − 4·29-s − 5·31-s + 32-s + 6·34-s + 8·35-s − 3·37-s + 3·38-s + 2·40-s + 6·41-s + 7·43-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s + 0.353·8-s + 0.632·10-s − 0.603·11-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.688·19-s + 0.447·20-s − 0.426·22-s − 0.834·23-s − 1/5·25-s + 0.755·28-s − 0.742·29-s − 0.898·31-s + 0.176·32-s + 1.02·34-s + 1.35·35-s − 0.493·37-s + 0.486·38-s + 0.316·40-s + 0.937·41-s + 1.06·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.247314245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.247314245\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.66211432600708658040654820546, −7.19313390637975459555865178762, −5.91878578282286308480002959718, −5.60967641771308961718745839468, −5.17898475978334414625003273760, −4.28174081831018311494637599975, −3.56193439914507813441186691228, −2.49095391521597970737328698000, −1.90402401559934755471830627394, −1.06570387643644874631526716438,
1.06570387643644874631526716438, 1.90402401559934755471830627394, 2.49095391521597970737328698000, 3.56193439914507813441186691228, 4.28174081831018311494637599975, 5.17898475978334414625003273760, 5.60967641771308961718745839468, 5.91878578282286308480002959718, 7.19313390637975459555865178762, 7.66211432600708658040654820546
