| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s − 5·11-s + 12-s + 13-s + 15-s + 16-s + 7·17-s − 18-s − 19-s + 20-s + 5·22-s − 8·23-s − 24-s + 25-s − 26-s + 27-s − 6·29-s − 30-s − 6·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s − 1.07·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.82652777743091, −12.20158243221264, −11.97697919524388, −11.14193266016824, −10.88744690453909, −10.23905523864980, −10.02315363988294, −9.670057150550134, −9.166934377711642, −8.535972301066530, −8.134431891304887, −7.861777661363372, −7.373228109597133, −6.982960965446244, −6.181851815773322, −5.755387608782009, −5.426259448040087, −4.889419577094410, −4.047628885436881, −3.557203235260270, −3.084031028885122, −2.525204049548941, −1.824376596010339, −1.689400234199743, −0.6885092384509064, 0,
0.6885092384509064, 1.689400234199743, 1.824376596010339, 2.525204049548941, 3.084031028885122, 3.557203235260270, 4.047628885436881, 4.889419577094410, 5.426259448040087, 5.755387608782009, 6.181851815773322, 6.982960965446244, 7.373228109597133, 7.861777661363372, 8.134431891304887, 8.535972301066530, 9.166934377711642, 9.670057150550134, 10.02315363988294, 10.23905523864980, 10.88744690453909, 11.14193266016824, 11.97697919524388, 12.20158243221264, 12.82652777743091
