| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 11-s + 2·13-s + 2·14-s + 16-s + 17-s − 19-s − 22-s + 6·23-s − 2·26-s − 2·28-s + 6·29-s − 2·31-s − 32-s − 34-s − 10·37-s + 38-s − 7·41-s + 8·43-s + 44-s − 6·46-s − 8·47-s − 3·49-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.301·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.229·19-s − 0.213·22-s + 1.25·23-s − 0.392·26-s − 0.377·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.171·34-s − 1.64·37-s + 0.162·38-s − 1.09·41-s + 1.21·43-s + 0.150·44-s − 0.884·46-s − 1.16·47-s − 3/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.24571678915138, −14.84886000509996, −14.09646688413419, −13.63711410161060, −13.05281567386205, −12.45670978439780, −12.09203125376025, −11.35121233603375, −10.94160137744099, −10.22515782965503, −10.00917580936238, −9.194190664972816, −8.788357452220075, −8.407344317630180, −7.589687494603809, −6.981016020454386, −6.606604693083922, −6.018820600839340, −5.308699852911917, −4.648941471486729, −3.706577880493583, −3.258391074370061, −2.581176857889181, −1.649022503105497, −0.9635405850591003, 0,
0.9635405850591003, 1.649022503105497, 2.581176857889181, 3.258391074370061, 3.706577880493583, 4.648941471486729, 5.308699852911917, 6.018820600839340, 6.606604693083922, 6.981016020454386, 7.589687494603809, 8.407344317630180, 8.788357452220075, 9.194190664972816, 10.00917580936238, 10.22515782965503, 10.94160137744099, 11.35121233603375, 12.09203125376025, 12.45670978439780, 13.05281567386205, 13.63711410161060, 14.09646688413419, 14.84886000509996, 15.24571678915138
