| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 3·9-s + 4·11-s + 2·13-s + 2·14-s + 16-s − 3·18-s + 8·19-s + 4·22-s + 6·23-s + 2·26-s + 2·28-s + 6·29-s − 10·31-s + 32-s − 3·36-s − 2·37-s + 8·38-s + 4·41-s − 8·43-s + 4·44-s + 6·46-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 − 9-s + 1.20·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.707·18-s + 1.83·19-s + 0.852·22-s + 1.25·23-s + 0.392·26-s + 0.377·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s − 1/2·36-s − 0.328·37-s + 1.29·38-s + 0.624·41-s − 1.21·43-s + 0.603·44-s + 0.884·46-s − 3/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.684524348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.684524348\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−16.20607114582959, −15.39844305741230, −14.75783964644995, −14.44823408029475, −13.95677390538870, −13.48213332700186, −12.77899200755569, −11.95056322674268, −11.70997469982272, −11.16391220519373, −10.74891185994230, −9.762673794808327, −9.145968469118401, −8.612376430400250, −7.984475897108632, −7.165699277644705, −6.732888504486613, −5.896621521132436, −5.295540590473126, −4.904436603062681, −3.863984106589646, −3.403098150505168, −2.656654789167849, −1.600702157170111, −0.9277233184648868,
0.9277233184648868, 1.600702157170111, 2.656654789167849, 3.403098150505168, 3.863984106589646, 4.904436603062681, 5.295540590473126, 5.896621521132436, 6.732888504486613, 7.165699277644705, 7.984475897108632, 8.612376430400250, 9.145968469118401, 9.762673794808327, 10.74891185994230, 11.16391220519373, 11.70997469982272, 11.95056322674268, 12.77899200755569, 13.48213332700186, 13.95677390538870, 14.44823408029475, 14.75783964644995, 15.39844305741230, 16.20607114582959
