| L(s) = 1 | − 2·3-s − 5-s + 7-s + 9-s − 2·11-s + 2·15-s − 3·17-s + 8·19-s − 2·21-s + 6·23-s − 4·25-s + 4·27-s + 9·29-s − 6·31-s + 4·33-s − 35-s − 3·37-s − 3·41-s + 2·43-s − 45-s − 12·47-s + 49-s + 6·51-s + 5·53-s + 2·55-s − 16·57-s + 14·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.516·15-s − 0.727·17-s + 1.83·19-s − 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.769·27-s + 1.67·29-s − 1.07·31-s + 0.696·33-s − 0.169·35-s − 0.493·37-s − 0.468·41-s + 0.304·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.686·53-s + 0.269·55-s − 2.11·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.339046485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.339046485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.12997794508243, −13.45186791270141, −12.97034615220516, −12.53099651956429, −11.75097876415397, −11.57705785395173, −11.29786857912548, −10.60595431872505, −10.20110656092184, −9.631445373898882, −8.971321796220456, −8.350172818520624, −7.980823072031512, −7.148229596589536, −6.938780560092628, −6.295405583529178, −5.507333048473361, −5.128706413041670, −4.922012532425480, −4.035256443845171, −3.382103338254500, −2.759954810016404, −1.952033621858420, −0.9962068225469121, −0.5047919084793981,
0.5047919084793981, 0.9962068225469121, 1.952033621858420, 2.759954810016404, 3.382103338254500, 4.035256443845171, 4.922012532425480, 5.128706413041670, 5.507333048473361, 6.295405583529178, 6.938780560092628, 7.148229596589536, 7.980823072031512, 8.350172818520624, 8.971321796220456, 9.631445373898882, 10.20110656092184, 10.60595431872505, 11.29786857912548, 11.57705785395173, 11.75097876415397, 12.53099651956429, 12.97034615220516, 13.45186791270141, 14.12997794508243
