| L(s) = 1 | + 7-s − 4·11-s − 13-s − 17-s − 5·19-s + 4·23-s + 3·29-s − 8·31-s − 4·37-s + 2·41-s + 7·47-s + 49-s + 9·53-s − 10·59-s − 13·61-s + 6·67-s + 8·71-s − 2·73-s − 4·77-s − 3·79-s + 12·83-s + 11·89-s − 91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s − 0.277·13-s − 0.242·17-s − 1.14·19-s + 0.834·23-s + 0.557·29-s − 1.43·31-s − 0.657·37-s + 0.312·41-s + 1.02·47-s + 1/7·49-s + 1.23·53-s − 1.30·59-s − 1.66·61-s + 0.733·67-s + 0.949·71-s − 0.234·73-s − 0.455·77-s − 0.337·79-s + 1.31·83-s + 1.16·89-s − 0.104·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300244194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300244194\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−12.75698490388612, −12.21672402463520, −11.89800893790694, −11.08119749940037, −10.81774952235419, −10.55897842203331, −10.06891306710860, −9.350541911540839, −8.967659171355189, −8.575634014605602, −8.037639634056223, −7.443014863876798, −7.313525520102144, −6.554452553963216, −6.105143838712786, −5.510858481897586, −5.037530013544819, −4.683893323777897, −4.081404300733649, −3.466943270592998, −2.892286699942497, −2.240051004749834, −1.973527929355364, −1.066741998736456, −0.3192894610442933,
0.3192894610442933, 1.066741998736456, 1.973527929355364, 2.240051004749834, 2.892286699942497, 3.466943270592998, 4.081404300733649, 4.683893323777897, 5.037530013544819, 5.510858481897586, 6.105143838712786, 6.554452553963216, 7.313525520102144, 7.443014863876798, 8.037639634056223, 8.575634014605602, 8.967659171355189, 9.350541911540839, 10.06891306710860, 10.55897842203331, 10.81774952235419, 11.08119749940037, 11.89800893790694, 12.21672402463520, 12.75698490388612
