| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s − 5·13-s + 2·15-s + 6·17-s + 4·19-s − 21-s − 23-s − 25-s + 27-s + 29-s + 31-s − 2·35-s + 12·37-s − 5·39-s + 9·41-s + 5·43-s + 2·45-s − 10·47-s + 49-s + 6·51-s − 2·53-s + 4·57-s − 13·59-s + 11·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.38·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 0.208·23-s − 1/5·25-s + 0.192·27-s + 0.185·29-s + 0.179·31-s − 0.338·35-s + 1.97·37-s − 0.800·39-s + 1.40·41-s + 0.762·43-s + 0.298·45-s − 1.45·47-s + 1/7·49-s + 0.840·51-s − 0.274·53-s + 0.529·57-s − 1.69·59-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.674977243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.674977243\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.70608023402108, −14.17227695133336, −13.92251859272383, −13.21046894517790, −12.72315659805028, −12.32120851935310, −11.73202521473413, −11.09344626521266, −10.31807791306317, −9.797719737862867, −9.506962310833391, −9.338300192048065, −8.159710925609919, −7.881147648023849, −7.324072754900727, −6.709964987694914, −5.885788647188117, −5.634092726689703, −4.840711558358780, −4.252033554795803, −3.382468391013634, −2.801953248252228, −2.340295978253592, −1.474695117198485, −0.6875047651436160,
0.6875047651436160, 1.474695117198485, 2.340295978253592, 2.801953248252228, 3.382468391013634, 4.252033554795803, 4.840711558358780, 5.634092726689703, 5.885788647188117, 6.709964987694914, 7.324072754900727, 7.881147648023849, 8.159710925609919, 9.338300192048065, 9.506962310833391, 9.797719737862867, 10.31807791306317, 11.09344626521266, 11.73202521473413, 12.32120851935310, 12.72315659805028, 13.21046894517790, 13.92251859272383, 14.17227695133336, 14.70608023402108
