| L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 4·11-s − 6·13-s + 4·15-s + 4·17-s − 4·19-s − 25-s + 4·27-s + 10·29-s + 2·31-s − 8·33-s + 37-s + 12·39-s + 2·41-s − 4·43-s − 2·45-s − 8·51-s − 6·53-s − 8·55-s + 8·57-s − 14·61-s + 12·65-s + 12·67-s + 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 1/5·25-s + 0.769·27-s + 1.85·29-s + 0.359·31-s − 1.39·33-s + 0.164·37-s + 1.92·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s − 1.12·51-s − 0.824·53-s − 1.07·55-s + 1.05·57-s − 1.79·61-s + 1.48·65-s + 1.46·67-s + 0.949·71-s + 1.63·73-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
| \(L(1)\) |
\(\approx\) |
\(0.8263531068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8263531068\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.73862430193905, −12.71919084010601, −12.39822233887746, −12.09177332543766, −11.83941189852528, −11.20715508369652, −10.90244612047908, −10.13427401332143, −9.868903156999647, −9.316101237393453, −8.531446929722035, −8.160687907347342, −7.585205577938615, −7.036929526179191, −6.424964590374182, −6.297653390297802, −5.392743893340687, −4.979690690162410, −4.414567044729325, −4.061245314898044, −3.209419087914663, −2.695791777191551, −1.801097608716552, −0.9655348562538716, −0.3710194423488552,
0.3710194423488552, 0.9655348562538716, 1.801097608716552, 2.695791777191551, 3.209419087914663, 4.061245314898044, 4.414567044729325, 4.979690690162410, 5.392743893340687, 6.297653390297802, 6.424964590374182, 7.036929526179191, 7.585205577938615, 8.160687907347342, 8.531446929722035, 9.316101237393453, 9.868903156999647, 10.13427401332143, 10.90244612047908, 11.20715508369652, 11.83941189852528, 12.09177332543766, 12.39822233887746, 12.71919084010601, 13.73862430193905
