| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 2·9-s − 12-s + 4·13-s + 14-s + 16-s − 2·18-s − 4·19-s − 21-s − 24-s + 4·26-s + 5·27-s + 28-s − 6·29-s − 10·31-s + 32-s − 2·36-s − 8·37-s − 4·38-s − 4·39-s − 3·41-s − 42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s − 0.917·19-s − 0.218·21-s − 0.204·24-s + 0.784·26-s + 0.962·27-s + 0.188·28-s − 1.11·29-s − 1.79·31-s + 0.176·32-s − 1/3·36-s − 1.31·37-s − 0.648·38-s − 0.640·39-s − 0.468·41-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.58782614872299659518826436146, −6.77225875616749292657705995963, −6.20620608322899979477353796201, −5.42481391160643228666592787913, −5.10520298878026508299683032031, −3.93053394658495608552520601032, −3.51638936968739224267517159742, −2.32852208034552551362593814035, −1.47232787133467714144933403293, 0,
1.47232787133467714144933403293, 2.32852208034552551362593814035, 3.51638936968739224267517159742, 3.93053394658495608552520601032, 5.10520298878026508299683032031, 5.42481391160643228666592787913, 6.20620608322899979477353796201, 6.77225875616749292657705995963, 7.58782614872299659518826436146
