| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 3·7-s − 8-s + 9-s − 2·10-s − 5·11-s − 12-s − 13-s + 3·14-s − 2·15-s + 16-s + 5·17-s − 18-s − 6·19-s + 2·20-s + 3·21-s + 5·22-s + 3·23-s + 24-s − 25-s + 26-s − 27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.50·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s − 0.516·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 1.37·19-s + 0.447·20-s + 0.654·21-s + 1.06·22-s + 0.625·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 474 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 474 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 79 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−10.26654239627389515631056683681, −9.947684152353270934723879199442, −8.967804168387287255389573412686, −7.79391119369772281647296286040, −6.84268861807480107704102527708, −5.91973300003215750552458540761, −5.19494737676757719955570680156, −3.31829273462319509512303304800, −2.03789622258225414220676544033, 0,
2.03789622258225414220676544033, 3.31829273462319509512303304800, 5.19494737676757719955570680156, 5.91973300003215750552458540761, 6.84268861807480107704102527708, 7.79391119369772281647296286040, 8.967804168387287255389573412686, 9.947684152353270934723879199442, 10.26654239627389515631056683681
