| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s + 13-s − 4·14-s + 16-s − 6·17-s − 4·19-s − 20-s + 6·23-s + 25-s + 26-s − 4·28-s − 6·29-s − 10·31-s + 32-s − 6·34-s + 4·35-s − 10·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s + 6·46-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 − 1.45·17-s − 0.917·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 1.11·29-s − 1.79·31-s + 0.176·32-s − 1.02·34-s + 0.676·35-s − 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.884·46-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−9.135950236841142485021394687069, −8.833501452594481304686000467623, −7.33631593435359451803501907108, −6.82690554790068919128090018298, −6.05406605235866086152380732766, −5.03841360881332404313911056034, −3.91297030993995478652961041358, −3.32886722585055639113607194217, −2.11797138782863684342786679274, 0,
2.11797138782863684342786679274, 3.32886722585055639113607194217, 3.91297030993995478652961041358, 5.03841360881332404313911056034, 6.05406605235866086152380732766, 6.82690554790068919128090018298, 7.33631593435359451803501907108, 8.833501452594481304686000467623, 9.135950236841142485021394687069
