| L(s) = 1 | − 2·5-s + 4·7-s + 2·13-s − 6·17-s + 4·19-s − 25-s + 2·29-s + 4·31-s − 8·35-s − 2·37-s + 2·41-s − 4·43-s − 8·47-s + 9·49-s − 10·53-s + 4·59-s − 6·61-s − 4·65-s + 4·67-s + 16·71-s + 6·73-s − 4·79-s + 12·83-s + 12·85-s − 10·89-s + 8·91-s − 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s + 0.718·31-s − 1.35·35-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + 1.89·71-s + 0.702·73-s − 0.450·79-s + 1.31·83-s + 1.30·85-s − 1.05·89-s + 0.838·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 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 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.26796454445225, −14.76507752469656, −14.15314171793565, −13.67196761489512, −13.29693308626416, −12.35246456759778, −12.04628740346188, −11.33247356415106, −11.14104573080138, −10.76788527212480, −9.809256060126891, −9.325225037732931, −8.525797756680857, −8.045513978164775, −7.989483937820556, −6.999428386540348, −6.643795238439823, −5.779850256736260, −5.012316245763178, −4.665273724916478, −4.036119934092327, −3.414795863819597, −2.537946314388652, −1.753723417876367, −1.074528620775553, 0,
1.074528620775553, 1.753723417876367, 2.537946314388652, 3.414795863819597, 4.036119934092327, 4.665273724916478, 5.012316245763178, 5.779850256736260, 6.643795238439823, 6.999428386540348, 7.989483937820556, 8.045513978164775, 8.525797756680857, 9.325225037732931, 9.809256060126891, 10.76788527212480, 11.14104573080138, 11.33247356415106, 12.04628740346188, 12.35246456759778, 13.29693308626416, 13.67196761489512, 14.15314171793565, 14.76507752469656, 15.26796454445225
