| L(s) = 1 | − 3·9-s + 3·11-s − 2·13-s + 6·17-s + 4·19-s + 2·23-s − 5·25-s + 3·29-s + 6·31-s − 3·37-s + 2·41-s − 6·43-s − 3·47-s − 13·53-s − 14·59-s + 15·61-s + 15·67-s + 5·71-s + 5·73-s + 79-s + 9·81-s + 5·83-s + 9·89-s − 13·97-s − 9·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 9-s + 0.904·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.417·23-s − 25-s + 0.557·29-s + 1.07·31-s − 0.493·37-s + 0.312·41-s − 0.914·43-s − 0.437·47-s − 1.78·53-s − 1.82·59-s + 1.92·61-s + 1.83·67-s + 0.593·71-s + 0.585·73-s + 0.112·79-s + 81-s + 0.548·83-s + 0.953·89-s − 1.31·97-s − 0.904·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.536598312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536598312\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−12.76870857018257, −12.19453309888845, −11.89576349893432, −11.71934798485023, −10.96715943769426, −10.70805178453331, −9.789855550947708, −9.580924180616551, −9.456003671610946, −8.411496719249737, −8.264295022249253, −7.801272128978853, −7.195427969252548, −6.587881710356108, −6.284458980942527, −5.587587945241334, −5.219183583275168, −4.782391344931566, −4.013523060443529, −3.408086388988240, −3.122987036583010, −2.483767319559710, −1.730264041642055, −1.105791289367148, −0.4776766422060637,
0.4776766422060637, 1.105791289367148, 1.730264041642055, 2.483767319559710, 3.122987036583010, 3.408086388988240, 4.013523060443529, 4.782391344931566, 5.219183583275168, 5.587587945241334, 6.284458980942527, 6.587881710356108, 7.195427969252548, 7.801272128978853, 8.264295022249253, 8.411496719249737, 9.456003671610946, 9.580924180616551, 9.789855550947708, 10.70805178453331, 10.96715943769426, 11.71934798485023, 11.89576349893432, 12.19453309888845, 12.76870857018257
