| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s + 4·14-s + 16-s − 6·17-s − 18-s + 4·19-s + 4·21-s + 24-s − 27-s − 4·28-s − 6·29-s − 8·31-s − 32-s + 6·34-s + 36-s + 2·37-s − 4·38-s + 6·41-s − 4·42-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.204·24-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.648·38-s + 0.937·41-s − 0.617·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 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 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 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.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−15.63419341575444, −15.46174667174345, −14.70803041368595, −13.90867786210219, −13.32383483435033, −12.79844027680818, −12.50429655192188, −11.68413136556548, −11.21081290834853, −10.72728087251998, −10.14902133063942, −9.469004409465197, −9.206283794352783, −8.716489344672148, −7.553974126520551, −7.374311025809978, −6.667574693436817, −6.099071546021984, −5.699308357019428, −4.834491586603662, −3.956941867760258, −3.399544233618051, −2.574731821651473, −1.849156494062333, −0.7286167505725691, 0,
0.7286167505725691, 1.849156494062333, 2.574731821651473, 3.399544233618051, 3.956941867760258, 4.834491586603662, 5.699308357019428, 6.099071546021984, 6.667574693436817, 7.374311025809978, 7.553974126520551, 8.716489344672148, 9.206283794352783, 9.469004409465197, 10.14902133063942, 10.72728087251998, 11.21081290834853, 11.68413136556548, 12.50429655192188, 12.79844027680818, 13.32383483435033, 13.90867786210219, 14.70803041368595, 15.46174667174345, 15.63419341575444
