| L(s) = 1 | − 3-s − 2·5-s + 9-s + 11-s + 2·13-s + 2·15-s + 6·17-s − 25-s − 27-s + 2·29-s − 33-s + 37-s − 2·39-s + 2·41-s − 2·45-s − 8·47-s − 7·49-s − 6·51-s − 10·53-s − 2·55-s − 12·59-s + 2·61-s − 4·65-s + 4·67-s + 8·71-s − 6·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.174·33-s + 0.164·37-s − 0.320·39-s + 0.312·41-s − 0.298·45-s − 1.16·47-s − 49-s − 0.840·51-s − 1.37·53-s − 0.269·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 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 |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.96649245985693, −15.57931970299436, −14.92735457003459, −14.33108654458431, −13.90862668421972, −13.05836029996491, −12.56168614414692, −12.10249160188920, −11.47026464312395, −11.21757460801705, −10.47907030988301, −9.870222688970412, −9.362830879267184, −8.541838774103123, −7.828812975168652, −7.692562558836796, −6.707192046576153, −6.254731092504112, −5.560775896496036, −4.871462051602638, −4.234511421801445, −3.514255998688017, −3.039968815460512, −1.741845791262163, −1.011871038178776, 0,
1.011871038178776, 1.741845791262163, 3.039968815460512, 3.514255998688017, 4.234511421801445, 4.871462051602638, 5.560775896496036, 6.254731092504112, 6.707192046576153, 7.692562558836796, 7.828812975168652, 8.541838774103123, 9.362830879267184, 9.870222688970412, 10.47907030988301, 11.21757460801705, 11.47026464312395, 12.10249160188920, 12.56168614414692, 13.05836029996491, 13.90862668421972, 14.33108654458431, 14.92735457003459, 15.57931970299436, 15.96649245985693
