| L(s) = 1 | − 5-s − 11-s + 4·13-s + 2·17-s + 19-s + 4·23-s + 25-s + 2·29-s − 4·31-s + 8·37-s + 6·41-s − 12·43-s + 8·47-s − 7·49-s − 4·53-s + 55-s − 12·59-s + 10·61-s − 4·65-s + 2·67-s − 12·71-s + 14·73-s − 12·79-s − 8·83-s − 2·85-s − 6·89-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.10·13-s + 0.485·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 1.31·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s − 49-s − 0.549·53-s + 0.134·55-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 0.244·67-s − 1.42·71-s + 1.63·73-s − 1.35·79-s − 0.878·83-s − 0.216·85-s − 0.635·89-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 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
−13.54901221347234, −13.00276945946786, −12.73294209807528, −12.19197111703726, −11.47227960991618, −11.28878498430507, −10.82183230055237, −10.26409469084009, −9.723221309959756, −9.242950990180992, −8.681195797526370, −8.236459590107524, −7.813724154664069, −7.212343156665556, −6.821980916439028, −5.996175431694161, −5.859173199111862, −5.014417848244965, −4.615864514788063, −3.961875201321334, −3.357539756709952, −3.009919745089998, −2.235888218514837, −1.387025372421008, −0.9225614662503773, 0,
0.9225614662503773, 1.387025372421008, 2.235888218514837, 3.009919745089998, 3.357539756709952, 3.961875201321334, 4.615864514788063, 5.014417848244965, 5.859173199111862, 5.996175431694161, 6.821980916439028, 7.212343156665556, 7.813724154664069, 8.236459590107524, 8.681195797526370, 9.242950990180992, 9.723221309959756, 10.26409469084009, 10.82183230055237, 11.28878498430507, 11.47227960991618, 12.19197111703726, 12.73294209807528, 13.00276945946786, 13.54901221347234
