| L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s − 4·7-s + 8-s + 9-s − 10-s + 11-s − 2·12-s + 2·13-s − 4·14-s + 2·15-s + 16-s + 2·17-s + 18-s − 20-s + 8·21-s + 22-s − 2·24-s + 25-s + 2·26-s + 4·27-s − 4·28-s − 4·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.223·20-s + 1.74·21-s + 0.213·22-s − 0.408·24-s + 1/5·25-s + 0.392·26-s + 0.769·27-s − 0.755·28-s − 0.742·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150590 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150590 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.39985298584705, −12.98645556232368, −12.51321316289952, −12.27042937001137, −11.78384478192902, −11.17631549287757, −10.88535659547220, −10.48137123455056, −9.809291153200718, −9.307214891879604, −8.903027592147902, −8.125338403120782, −7.479842448519346, −7.026629407370767, −6.555985047955412, −6.119550347668192, −5.616710798081891, −5.374242774550079, −4.541679482615424, −4.003591150790264, −3.440419518565349, −3.161709910077342, −2.309046368314419, −1.462189782806119, −0.6581291492391769, 0,
0.6581291492391769, 1.462189782806119, 2.309046368314419, 3.161709910077342, 3.440419518565349, 4.003591150790264, 4.541679482615424, 5.374242774550079, 5.616710798081891, 6.119550347668192, 6.555985047955412, 7.026629407370767, 7.479842448519346, 8.125338403120782, 8.903027592147902, 9.307214891879604, 9.809291153200718, 10.48137123455056, 10.88535659547220, 11.17631549287757, 11.78384478192902, 12.27042937001137, 12.51321316289952, 12.98645556232368, 13.39985298584705
