| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s − 11-s + 2·14-s + 16-s + 2·17-s − 20-s − 22-s − 8·23-s + 25-s + 2·28-s + 8·29-s − 6·31-s + 32-s + 2·34-s − 2·35-s − 40-s + 10·41-s − 8·43-s − 44-s − 8·46-s − 3·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.377·28-s + 1.48·29-s − 1.07·31-s + 0.176·32-s + 0.342·34-s − 0.338·35-s − 0.158·40-s + 1.56·41-s − 1.21·43-s − 0.150·44-s − 1.17·46-s − 3/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.53167026893541, −12.94769795247677, −12.46371718203891, −12.11110369476223, −11.63141396985808, −11.26550747033980, −10.69938241720083, −10.23393423121506, −9.861709458953808, −9.068161423641625, −8.594703404079580, −7.914307236157718, −7.784144996940682, −7.257663840572081, −6.488092157584274, −6.114866441742409, −5.542506225772711, −4.973556609381825, −4.541547408830375, −4.048807118823869, −3.455607634069261, −2.916815228403990, −2.197709732348468, −1.686105965128623, −0.9286670414929944, 0,
0.9286670414929944, 1.686105965128623, 2.197709732348468, 2.916815228403990, 3.455607634069261, 4.048807118823869, 4.541547408830375, 4.973556609381825, 5.542506225772711, 6.114866441742409, 6.488092157584274, 7.257663840572081, 7.784144996940682, 7.914307236157718, 8.594703404079580, 9.068161423641625, 9.861709458953808, 10.23393423121506, 10.69938241720083, 11.26550747033980, 11.63141396985808, 12.11110369476223, 12.46371718203891, 12.94769795247677, 13.53167026893541
