| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s − 2·9-s + 12-s + 5·13-s − 4·14-s + 16-s − 3·17-s − 2·18-s − 19-s − 4·21-s + 3·23-s + 24-s − 5·25-s + 5·26-s − 5·27-s − 4·28-s + 9·29-s − 4·31-s + 32-s − 3·34-s − 2·36-s + 5·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s + 1.38·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.872·21-s + 0.625·23-s + 0.204·24-s − 25-s + 0.980·26-s − 0.962·27-s − 0.755·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s − 1/3·36-s + 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.524798432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524798432\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 53 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−13.48097305287934545041204893811, −13.15344609438075689944845079286, −11.81761494303416970928767268639, −10.69149667428673846399677584241, −9.381217023375044891193605681246, −8.357926167105308090415107881051, −6.72640894491877368549073868160, −5.85130331497128657353090562352, −3.90015209017905831393374696535, −2.81073095887439436341819860040,
2.81073095887439436341819860040, 3.90015209017905831393374696535, 5.85130331497128657353090562352, 6.72640894491877368549073868160, 8.357926167105308090415107881051, 9.381217023375044891193605681246, 10.69149667428673846399677584241, 11.81761494303416970928767268639, 13.15344609438075689944845079286, 13.48097305287934545041204893811
