| L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s + 8-s + 9-s + 4·10-s + 11-s − 12-s − 4·13-s − 4·15-s + 16-s + 2·17-s + 18-s + 4·20-s + 22-s − 6·23-s − 24-s + 11·25-s − 4·26-s − 27-s + 10·29-s − 4·30-s + 8·31-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s − 1.03·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.894·20-s + 0.213·22-s − 1.25·23-s − 0.204·24-s + 11/5·25-s − 0.784·26-s − 0.192·27-s + 1.85·29-s − 0.730·30-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.553922854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.553922854\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.734399327056068821213499579504, −7.74249612429138712743707138359, −6.69373326676767816887250497086, −6.35062410158064670794655966913, −5.56849277330524351488966193226, −5.01244554431640501263959251141, −4.22775784486927931972933466181, −2.84811781912171505509456779453, −2.18727745386797108535245912439, −1.12176628667720465571658579928,
1.12176628667720465571658579928, 2.18727745386797108535245912439, 2.84811781912171505509456779453, 4.22775784486927931972933466181, 5.01244554431640501263959251141, 5.56849277330524351488966193226, 6.35062410158064670794655966913, 6.69373326676767816887250497086, 7.74249612429138712743707138359, 8.734399327056068821213499579504
