| L(s) = 1 | − 5-s − 5·7-s + 2·11-s + 13-s + 3·17-s − 2·19-s + 4·23-s − 4·25-s − 6·29-s + 4·31-s + 5·35-s − 11·37-s − 8·41-s − 43-s + 9·47-s + 18·49-s − 12·53-s − 2·55-s − 6·59-s − 65-s + 6·67-s + 7·71-s − 2·73-s − 10·77-s − 12·79-s + 16·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s + 0.603·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.834·23-s − 4/5·25-s − 1.11·29-s + 0.718·31-s + 0.845·35-s − 1.80·37-s − 1.24·41-s − 0.152·43-s + 1.31·47-s + 18/7·49-s − 1.64·53-s − 0.269·55-s − 0.781·59-s − 0.124·65-s + 0.733·67-s + 0.830·71-s − 0.234·73-s − 1.13·77-s − 1.35·79-s + 1.75·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9664144548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9664144548\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−7.82172208330940216022728438102, −7.04394008714092547161854547638, −6.57128991023068628933570462847, −5.93902269695216238609503453080, −5.16509117765532653048329281952, −4.04372769929878060663009831205, −3.51318396517850651773017534778, −2.99263455242250842514678831679, −1.76199942492284124004695187158, −0.48253138144269349132738130603,
0.48253138144269349132738130603, 1.76199942492284124004695187158, 2.99263455242250842514678831679, 3.51318396517850651773017534778, 4.04372769929878060663009831205, 5.16509117765532653048329281952, 5.93902269695216238609503453080, 6.57128991023068628933570462847, 7.04394008714092547161854547638, 7.82172208330940216022728438102
