| L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 2·17-s − 4·19-s − 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s − 2·45-s − 7·49-s − 2·51-s − 2·53-s − 8·55-s + 4·57-s + 4·59-s − 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s − 49-s − 0.280·51-s − 0.274·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5391289118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5391289118\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−17.67075195000178004295107787310, −16.62726100086658933765837426594, −15.47325164750050957705309114011, −14.18198826155077042068868714860, −12.31725686074733747032880351584, −11.58204746144152243175303945055, −9.964671915727173343349228861429, −8.098990694093691505068092710868, −6.42897107072744719896577758454, −4.25303028692796488061253338187,
4.25303028692796488061253338187, 6.42897107072744719896577758454, 8.098990694093691505068092710868, 9.964671915727173343349228861429, 11.58204746144152243175303945055, 12.31725686074733747032880351584, 14.18198826155077042068868714860, 15.47325164750050957705309114011, 16.62726100086658933765837426594, 17.67075195000178004295107787310
