| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s + 2·13-s + 16-s + 4·19-s + 6·22-s + 5·23-s + 2·26-s + 2·31-s + 32-s + 3·37-s + 4·38-s + 5·41-s − 2·43-s + 6·44-s + 5·46-s − 7·49-s + 2·52-s + 53-s + 3·59-s − 5·61-s + 2·62-s + 64-s + 2·67-s + 5·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 0.554·13-s + 1/4·16-s + 0.917·19-s + 1.27·22-s + 1.04·23-s + 0.392·26-s + 0.359·31-s + 0.176·32-s + 0.493·37-s + 0.648·38-s + 0.780·41-s − 0.304·43-s + 0.904·44-s + 0.737·46-s − 49-s + 0.277·52-s + 0.137·53-s + 0.390·59-s − 0.640·61-s + 0.254·62-s + 1/8·64-s + 0.244·67-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.768477916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.768477916\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.60748304606569, −12.94185401277792, −12.62002689144911, −12.00904671727965, −11.53655279417387, −11.31541407904173, −10.78862014740687, −10.08804498472945, −9.537253619686532, −9.173079372282950, −8.654837381306995, −8.043361072573932, −7.423873206916997, −6.940223356063052, −6.408062342496272, −6.115882414648645, −5.408608243040304, −4.873296011691316, −4.296505457045605, −3.781297443916336, −3.298655106731629, −2.752561999621320, −1.875158812821216, −1.252362176953162, −0.7711326598259453,
0.7711326598259453, 1.252362176953162, 1.875158812821216, 2.752561999621320, 3.298655106731629, 3.781297443916336, 4.296505457045605, 4.873296011691316, 5.408608243040304, 6.115882414648645, 6.408062342496272, 6.940223356063052, 7.423873206916997, 8.043361072573932, 8.654837381306995, 9.173079372282950, 9.537253619686532, 10.08804498472945, 10.78862014740687, 11.31541407904173, 11.53655279417387, 12.00904671727965, 12.62002689144911, 12.94185401277792, 13.60748304606569
