| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 2·13-s − 15-s + 6·17-s − 8·19-s − 4·21-s + 25-s + 27-s + 2·29-s − 8·31-s + 4·33-s + 4·35-s − 2·37-s − 2·39-s + 2·41-s − 43-s − 45-s + 8·47-s + 9·49-s + 6·51-s + 6·53-s − 4·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.152·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.676666724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676666724\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.376108195663805561321528729807, −7.32931519698551785382460092653, −6.91137859860674683865734955347, −6.19894764455387152020699847117, −5.39580068536724069843640364007, −4.09679395742503110976688839048, −3.79403924818835144177183627000, −2.97520595717042768646891191084, −2.03333223410725860124570841177, −0.66759142542773483292323325865,
0.66759142542773483292323325865, 2.03333223410725860124570841177, 2.97520595717042768646891191084, 3.79403924818835144177183627000, 4.09679395742503110976688839048, 5.39580068536724069843640364007, 6.19894764455387152020699847117, 6.91137859860674683865734955347, 7.32931519698551785382460092653, 8.376108195663805561321528729807
