| L(s) = 1 | − 5·7-s + 2·13-s + 19-s − 11·31-s + 11·37-s + 13·43-s + 18·49-s − 13·61-s + 16·67-s − 7·73-s − 17·79-s − 10·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 0.554·13-s + 0.229·19-s − 1.97·31-s + 1.80·37-s + 1.98·43-s + 18/7·49-s − 1.66·61-s + 1.95·67-s − 0.819·73-s − 1.91·79-s − 1.04·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.69294284815195, −16.20331783857815, −15.79327225668184, −15.28004063107701, −14.44684551632988, −13.96839503262676, −13.17088769232611, −12.80615705021284, −12.51280577063326, −11.53940244872834, −11.02821292883188, −10.31039188890981, −9.767101353613159, −9.122869852592304, −8.870864068014507, −7.676113999785867, −7.316526357301291, −6.440161177218909, −6.026353390290050, −5.447257820643474, −4.292701666896565, −3.711053992663052, −3.044035501108180, −2.321072132321339, −1.039909908546452, 0,
1.039909908546452, 2.321072132321339, 3.044035501108180, 3.711053992663052, 4.292701666896565, 5.447257820643474, 6.026353390290050, 6.440161177218909, 7.316526357301291, 7.676113999785867, 8.870864068014507, 9.122869852592304, 9.767101353613159, 10.31039188890981, 11.02821292883188, 11.53940244872834, 12.51280577063326, 12.80615705021284, 13.17088769232611, 13.96839503262676, 14.44684551632988, 15.28004063107701, 15.79327225668184, 16.20331783857815, 16.69294284815195
