| L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s − 13-s + 4·14-s + 16-s + 2·17-s + 4·19-s + 20-s − 8·23-s + 25-s − 26-s + 4·28-s − 2·29-s − 8·31-s + 32-s + 2·34-s + 4·35-s + 2·37-s + 4·38-s + 40-s + 6·41-s + 12·43-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.676·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.82·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.214245932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.214245932\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.839542125076117174938666696331, −8.998292228580138090371311251918, −7.70002191137841568751282880154, −7.61942621520258683571159755696, −6.12276299001965025887943314435, −5.47980176854220799955648347826, −4.68667184047043357988258422645, −3.75778333243500449289895503228, −2.40122970806064783978264870635, −1.46269795593787510909097790861,
1.46269795593787510909097790861, 2.40122970806064783978264870635, 3.75778333243500449289895503228, 4.68667184047043357988258422645, 5.47980176854220799955648347826, 6.12276299001965025887943314435, 7.61942621520258683571159755696, 7.70002191137841568751282880154, 8.998292228580138090371311251918, 9.839542125076117174938666696331
