| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 3·9-s − 10-s − 5·11-s + 2·13-s + 16-s + 6·17-s − 3·18-s − 19-s − 20-s − 5·22-s − 5·23-s − 4·25-s + 2·26-s − 4·31-s + 32-s + 6·34-s − 3·36-s − 4·37-s − 38-s − 40-s − 4·41-s − 11·43-s − 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 9-s − 0.316·10-s − 1.50·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.229·19-s − 0.223·20-s − 1.06·22-s − 1.04·23-s − 4/5·25-s + 0.392·26-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 1/2·36-s − 0.657·37-s − 0.162·38-s − 0.158·40-s − 0.624·41-s − 1.67·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−8.533996676394008024675283283549, −8.016528076809825022150973392628, −7.38198996205643046296223203251, −6.15902403346894630494770621050, −5.57912843528416519108890141545, −4.88186262248798429592092971654, −3.63635230249452893543488543793, −3.09289003342780337303183300496, −1.90813661718622831799099531234, 0,
1.90813661718622831799099531234, 3.09289003342780337303183300496, 3.63635230249452893543488543793, 4.88186262248798429592092971654, 5.57912843528416519108890141545, 6.15902403346894630494770621050, 7.38198996205643046296223203251, 8.016528076809825022150973392628, 8.533996676394008024675283283549
