| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 2·11-s + 12-s − 13-s − 14-s + 16-s − 6·17-s + 18-s + 4·19-s − 21-s + 2·22-s + 2·23-s + 24-s − 26-s + 27-s − 28-s + 6·29-s + 8·31-s + 32-s + 2·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.426·22-s + 0.417·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.348·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.462273022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.462273022\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−15.98831598011956, −15.49020783932526, −15.16167469608536, −14.30455722774342, −14.01581859009700, −13.39568362282779, −13.01668006204712, −12.21823251509567, −11.82565803322438, −11.19496832566536, −10.47588307341493, −9.838553217218854, −9.325592410184907, −8.563054914728343, −8.127470980441212, −7.140174643258967, −6.735720812165930, −6.266326254809653, −5.229046555776300, −4.699004178714773, −4.008928989580082, −3.275585124621611, −2.665881642063684, −1.884357194525093, −0.8192219115488724,
0.8192219115488724, 1.884357194525093, 2.665881642063684, 3.275585124621611, 4.008928989580082, 4.699004178714773, 5.229046555776300, 6.266326254809653, 6.735720812165930, 7.140174643258967, 8.127470980441212, 8.563054914728343, 9.325592410184907, 9.838553217218854, 10.47588307341493, 11.19496832566536, 11.82565803322438, 12.21823251509567, 13.01668006204712, 13.39568362282779, 14.01581859009700, 14.30455722774342, 15.16167469608536, 15.49020783932526, 15.98831598011956
