| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 2·7-s + 9-s + 2·10-s + 4·11-s − 2·12-s + 4·14-s − 15-s − 4·16-s + 2·17-s + 2·18-s − 2·19-s + 2·20-s − 2·21-s + 8·22-s − 8·23-s − 4·25-s − 27-s + 4·28-s − 3·29-s − 2·30-s − 31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s + 1.06·14-s − 0.258·15-s − 16-s + 0.485·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s − 0.436·21-s + 1.70·22-s − 1.66·23-s − 4/5·25-s − 0.192·27-s + 0.755·28-s − 0.557·29-s − 0.365·30-s − 0.179·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.412005156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.412005156\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 97 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
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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
−11.85034881237549382977948120603, −11.38485172495255005565280976546, −10.12608990331471313754460294414, −9.042406233064135325169152837469, −7.63941991487258172879610952903, −6.29147486031868245163748076757, −5.78120113549963226756648321198, −4.60343669826099046651925767517, −3.77583287923510274229114744932, −1.93096928370614500872650074487,
1.93096928370614500872650074487, 3.77583287923510274229114744932, 4.60343669826099046651925767517, 5.78120113549963226756648321198, 6.29147486031868245163748076757, 7.63941991487258172879610952903, 9.042406233064135325169152837469, 10.12608990331471313754460294414, 11.38485172495255005565280976546, 11.85034881237549382977948120603
