| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4·11-s + 15-s + 8·17-s + 4·19-s − 21-s + 6·23-s + 25-s − 27-s + 10·29-s + 4·31-s + 4·33-s − 35-s + 6·37-s − 6·41-s − 8·43-s − 45-s − 6·47-s + 49-s − 8·51-s − 6·53-s + 4·55-s − 4·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.258·15-s + 1.94·17-s + 0.917·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.696·33-s − 0.169·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 1.12·51-s − 0.824·53-s + 0.539·55-s − 0.529·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 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 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.40724543459673, −13.78596636163233, −13.42514104953183, −12.72378485899416, −12.34461560836018, −11.83919312018030, −11.44217538278989, −10.92360939137262, −10.21113962998796, −10.06086581805466, −9.489946674183644, −8.566128509994443, −8.114256149490665, −7.797489191481844, −7.186445168142804, −6.641176844758415, −5.983952564049869, −5.219355584565556, −5.069895595118919, −4.540702829180498, −3.565870046139563, −3.048902572103223, −2.596359316034745, −1.303055440421916, −1.040077964611272, 0,
1.040077964611272, 1.303055440421916, 2.596359316034745, 3.048902572103223, 3.565870046139563, 4.540702829180498, 5.069895595118919, 5.219355584565556, 5.983952564049869, 6.641176844758415, 7.186445168142804, 7.797489191481844, 8.114256149490665, 8.566128509994443, 9.489946674183644, 10.06086581805466, 10.21113962998796, 10.92360939137262, 11.44217538278989, 11.83919312018030, 12.34461560836018, 12.72378485899416, 13.42514104953183, 13.78596636163233, 14.40724543459673
