| L(s) = 1 | − 5-s + 7-s − 2·11-s − 17-s + 6·19-s − 8·23-s − 4·25-s − 29-s − 4·31-s − 35-s + 3·37-s − 3·41-s − 2·43-s + 6·47-s + 49-s − 9·53-s + 2·55-s + 10·59-s − 61-s + 12·67-s + 2·71-s − 73-s − 2·77-s − 2·79-s + 6·83-s + 85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.603·11-s − 0.242·17-s + 1.37·19-s − 1.66·23-s − 4/5·25-s − 0.185·29-s − 0.718·31-s − 0.169·35-s + 0.493·37-s − 0.468·41-s − 0.304·43-s + 0.875·47-s + 1/7·49-s − 1.23·53-s + 0.269·55-s + 1.30·59-s − 0.128·61-s + 1.46·67-s + 0.237·71-s − 0.117·73-s − 0.227·77-s − 0.225·79-s + 0.658·83-s + 0.108·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107164268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107164268\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.39548579434277, −12.15337776607764, −11.59104519830790, −11.33210118377600, −10.83727591357792, −10.24933796860176, −9.867124984619795, −9.440725887039610, −8.955071894227455, −8.232459198617773, −7.897185985454762, −7.744218798612474, −7.034716304769378, −6.629525319262561, −5.915221549122897, −5.431927244691516, −5.219763116789616, −4.390044518844127, −4.022073172985572, −3.527235914159131, −2.934806240288423, −2.252423235023512, −1.823178321015339, −1.079477048980273, −0.2905920803478950,
0.2905920803478950, 1.079477048980273, 1.823178321015339, 2.252423235023512, 2.934806240288423, 3.527235914159131, 4.022073172985572, 4.390044518844127, 5.219763116789616, 5.431927244691516, 5.915221549122897, 6.629525319262561, 7.034716304769378, 7.744218798612474, 7.897185985454762, 8.232459198617773, 8.955071894227455, 9.440725887039610, 9.867124984619795, 10.24933796860176, 10.83727591357792, 11.33210118377600, 11.59104519830790, 12.15337776607764, 12.39548579434277
