| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 6·13-s − 15-s + 2·17-s − 4·19-s + 21-s + 4·23-s + 25-s + 27-s − 6·29-s − 35-s + 6·37-s − 6·39-s + 2·41-s + 4·43-s − 45-s + 8·47-s + 49-s + 2·51-s − 2·53-s − 4·57-s − 12·59-s − 6·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.169·35-s + 0.986·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 0.529·57-s − 1.56·59-s − 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.20651200445157, −13.35541685610331, −13.08205088731009, −12.49083073862810, −12.05540700952131, −11.70358882572150, −10.82566263301922, −10.70334828483103, −10.02630373906378, −9.359914940409862, −9.111933928864796, −8.579560351371553, −7.727630773466404, −7.561124752851898, −7.294260877000290, −6.383320762806826, −5.908885332378222, −5.118485438787180, −4.639439315159841, −4.256466514835704, −3.506280979746661, −2.888251981265848, −2.358846893908324, −1.759606628100814, −0.8616579133198362, 0,
0.8616579133198362, 1.759606628100814, 2.358846893908324, 2.888251981265848, 3.506280979746661, 4.256466514835704, 4.639439315159841, 5.118485438787180, 5.908885332378222, 6.383320762806826, 7.294260877000290, 7.561124752851898, 7.727630773466404, 8.579560351371553, 9.111933928864796, 9.359914940409862, 10.02630373906378, 10.70334828483103, 10.82566263301922, 11.70358882572150, 12.05540700952131, 12.49083073862810, 13.08205088731009, 13.35541685610331, 14.20651200445157
