| L(s) = 1 | − 2-s + 4-s − 8-s + 6·11-s − 5·13-s + 16-s − 6·17-s + 4·19-s − 6·22-s + 6·23-s − 5·25-s + 5·26-s + 6·29-s + 31-s − 32-s + 6·34-s − 37-s − 4·38-s + 6·41-s − 43-s + 6·44-s − 6·46-s + 6·47-s + 5·50-s − 5·52-s − 6·53-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s − 1.38·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.27·22-s + 1.25·23-s − 25-s + 0.980·26-s + 1.11·29-s + 0.179·31-s − 0.176·32-s + 1.02·34-s − 0.164·37-s − 0.648·38-s + 0.937·41-s − 0.152·43-s + 0.904·44-s − 0.884·46-s + 0.875·47-s + 0.707·50-s − 0.693·52-s − 0.824·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.306615511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306615511\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−9.057732301440674521981200930486, −8.213222746268842791227564813953, −7.17756521662208616357912102378, −6.86373819824463860920029366770, −5.99293789071915916490293453418, −4.87602663790637391350842024269, −4.09964776417661011092906723802, −2.95453084311476587817392272476, −1.97786137008912495989390313194, −0.807787551671847026691110476191,
0.807787551671847026691110476191, 1.97786137008912495989390313194, 2.95453084311476587817392272476, 4.09964776417661011092906723802, 4.87602663790637391350842024269, 5.99293789071915916490293453418, 6.86373819824463860920029366770, 7.17756521662208616357912102378, 8.213222746268842791227564813953, 9.057732301440674521981200930486
