| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s + 13-s + 2·15-s − 2·17-s + 4·19-s − 25-s − 27-s + 2·29-s + 4·33-s + 2·37-s − 39-s + 6·41-s + 4·43-s − 2·45-s − 8·47-s + 2·51-s + 10·53-s + 8·55-s − 4·57-s + 4·59-s − 2·61-s − 2·65-s − 4·67-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 0.280·51-s + 1.37·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.034919576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034919576\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 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 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.50767927408442, −13.04622273287353, −12.50481384262920, −12.06316321474279, −11.58015910827378, −11.16840005156831, −10.77006333990519, −10.21735883585554, −9.761160903134983, −9.163940897986748, −8.520774007252112, −8.064938708540131, −7.534707307895994, −7.266850035260321, −6.555798012911618, −5.969703853954974, −5.456568841278317, −4.969710441685193, −4.358842254632306, −3.917841703995996, −3.190903613041147, −2.658133529937827, −1.926832447134213, −0.9979049025063089, −0.3815350931011748,
0.3815350931011748, 0.9979049025063089, 1.926832447134213, 2.658133529937827, 3.190903613041147, 3.917841703995996, 4.358842254632306, 4.969710441685193, 5.456568841278317, 5.969703853954974, 6.555798012911618, 7.266850035260321, 7.534707307895994, 8.064938708540131, 8.520774007252112, 9.163940897986748, 9.761160903134983, 10.21735883585554, 10.77006333990519, 11.16840005156831, 11.58015910827378, 12.06316321474279, 12.50481384262920, 13.04622273287353, 13.50767927408442
