| L(s) = 1 | − 5-s − 3·7-s + 6·11-s − 6·17-s − 2·23-s + 25-s − 8·29-s − 3·31-s + 3·35-s − 6·37-s + 2·41-s + 11·43-s + 12·47-s + 2·49-s + 12·53-s − 6·55-s + 15·61-s + 5·67-s − 8·71-s + 5·73-s − 18·77-s − 15·79-s − 8·83-s + 6·85-s − 6·89-s + 97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.80·11-s − 1.45·17-s − 0.417·23-s + 1/5·25-s − 1.48·29-s − 0.538·31-s + 0.507·35-s − 0.986·37-s + 0.312·41-s + 1.67·43-s + 1.75·47-s + 2/7·49-s + 1.64·53-s − 0.809·55-s + 1.92·61-s + 0.610·67-s − 0.949·71-s + 0.585·73-s − 2.05·77-s − 1.68·79-s − 0.878·83-s + 0.650·85-s − 0.635·89-s + 0.101·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−15.50444755579228, −14.86031755606210, −14.30799919671729, −13.88117303751555, −13.13048924920842, −12.78870855496453, −12.21086538692929, −11.62393085651247, −11.23691760066036, −10.60900678225736, −9.925103348684182, −9.303263348768552, −8.945411573557880, −8.583964400509585, −7.520585009787014, −6.970067488195117, −6.745152208324761, −5.912964959311344, −5.538658366928315, −4.310202346927237, −4.000480775144036, −3.587697188098660, −2.605585264812545, −1.927953912285276, −0.9055067030351425, 0,
0.9055067030351425, 1.927953912285276, 2.605585264812545, 3.587697188098660, 4.000480775144036, 4.310202346927237, 5.538658366928315, 5.912964959311344, 6.745152208324761, 6.970067488195117, 7.520585009787014, 8.583964400509585, 8.945411573557880, 9.303263348768552, 9.925103348684182, 10.60900678225736, 11.23691760066036, 11.62393085651247, 12.21086538692929, 12.78870855496453, 13.13048924920842, 13.88117303751555, 14.30799919671729, 14.86031755606210, 15.50444755579228
