| L(s) = 1 | − 2·3-s + 3·5-s + 9-s − 11-s + 3·13-s − 6·15-s − 2·17-s − 4·19-s − 2·23-s + 4·25-s + 4·27-s − 2·29-s − 31-s + 2·33-s − 37-s − 6·39-s − 6·41-s − 2·43-s + 3·45-s + 12·47-s + 4·51-s − 9·53-s − 3·55-s + 8·57-s − 59-s − 2·61-s + 9·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 1.54·15-s − 0.485·17-s − 0.917·19-s − 0.417·23-s + 4/5·25-s + 0.769·27-s − 0.371·29-s − 0.179·31-s + 0.348·33-s − 0.164·37-s − 0.960·39-s − 0.937·41-s − 0.304·43-s + 0.447·45-s + 1.75·47-s + 0.560·51-s − 1.23·53-s − 0.404·55-s + 1.05·57-s − 0.130·59-s − 0.256·61-s + 1.11·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330922538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330922538\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−13.54816874970945, −13.17307353285882, −12.53539819148340, −12.27702306143487, −11.54960908924331, −11.10431052351956, −10.65408207259703, −10.38138904338354, −9.821331355898317, −9.173447316730359, −8.835584499128408, −8.215040732554127, −7.614485103519225, −6.778557748954734, −6.368717149620618, −6.176403712774226, −5.439311637665226, −5.271837883126009, −4.543569328006299, −3.929276856902814, −3.187936762221647, −2.362062116698241, −1.922739020557470, −1.227656700217921, −0.3903816168592632,
0.3903816168592632, 1.227656700217921, 1.922739020557470, 2.362062116698241, 3.187936762221647, 3.929276856902814, 4.543569328006299, 5.271837883126009, 5.439311637665226, 6.176403712774226, 6.368717149620618, 6.778557748954734, 7.614485103519225, 8.215040732554127, 8.835584499128408, 9.173447316730359, 9.821331355898317, 10.38138904338354, 10.65408207259703, 11.10431052351956, 11.54960908924331, 12.27702306143487, 12.53539819148340, 13.17307353285882, 13.54816874970945
