| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 4·11-s + 13-s − 2·14-s + 16-s − 4·17-s + 6·19-s − 4·22-s + 26-s − 2·28-s + 4·29-s + 6·31-s + 32-s − 4·34-s + 2·37-s + 6·38-s − 10·41-s − 8·43-s − 4·44-s − 3·49-s + 52-s − 4·53-s − 2·56-s + 4·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 1.20·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.852·22-s + 0.196·26-s − 0.377·28-s + 0.742·29-s + 1.07·31-s + 0.176·32-s − 0.685·34-s + 0.328·37-s + 0.973·38-s − 1.56·41-s − 1.21·43-s − 0.603·44-s − 3/7·49-s + 0.138·52-s − 0.549·53-s − 0.267·56-s + 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 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.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.72384123989891528047640757451, −6.76282149578399467772700679730, −6.43514663138306386742952238623, −5.43122093573879899350689892090, −4.95713876909416113101518859785, −4.08632407521995038572091105928, −3.06803172076760562588500516876, −2.74926699637595623228189559131, −1.47068428933794101158964544573, 0,
1.47068428933794101158964544573, 2.74926699637595623228189559131, 3.06803172076760562588500516876, 4.08632407521995038572091105928, 4.95713876909416113101518859785, 5.43122093573879899350689892090, 6.43514663138306386742952238623, 6.76282149578399467772700679730, 7.72384123989891528047640757451
