| L(s) = 1 | − 5-s + 7-s − 11-s + 13-s + 4·17-s + 19-s + 7·23-s + 25-s + 8·29-s − 9·31-s − 35-s + 12·37-s − 6·41-s − 5·43-s + 12·47-s + 49-s − 11·53-s + 55-s − 9·59-s + 5·61-s − 65-s + 71-s + 73-s − 77-s + 8·79-s − 15·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.277·13-s + 0.970·17-s + 0.229·19-s + 1.45·23-s + 1/5·25-s + 1.48·29-s − 1.61·31-s − 0.169·35-s + 1.97·37-s − 0.937·41-s − 0.762·43-s + 1.75·47-s + 1/7·49-s − 1.51·53-s + 0.134·55-s − 1.17·59-s + 0.640·61-s − 0.124·65-s + 0.118·71-s + 0.117·73-s − 0.113·77-s + 0.900·79-s − 1.64·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.15796449452401, −13.59842078682476, −12.91230915701697, −12.63977435524053, −12.13478050468138, −11.46268383583622, −11.17556803826575, −10.69504485212974, −10.11415046366560, −9.600682554982883, −9.021486621990550, −8.514322152500299, −8.043974696893221, −7.443737443534635, −7.188513904453350, −6.382943642811987, −5.914486358574740, −5.121028155433013, −4.931427316687540, −4.152657817003058, −3.561657739657646, −2.957804464461230, −2.472164287398032, −1.384381912977826, −1.035280872921251, 0,
1.035280872921251, 1.384381912977826, 2.472164287398032, 2.957804464461230, 3.561657739657646, 4.152657817003058, 4.931427316687540, 5.121028155433013, 5.914486358574740, 6.382943642811987, 7.188513904453350, 7.443737443534635, 8.043974696893221, 8.514322152500299, 9.021486621990550, 9.600682554982883, 10.11415046366560, 10.69504485212974, 11.17556803826575, 11.46268383583622, 12.13478050468138, 12.63977435524053, 12.91230915701697, 13.59842078682476, 14.15796449452401
