| L(s) = 1 | − 7-s − 6·11-s − 13-s − 8·17-s − 6·29-s + 2·31-s − 10·37-s + 8·41-s − 6·43-s − 8·47-s + 49-s − 12·53-s − 8·59-s + 6·61-s + 8·67-s + 4·71-s − 2·73-s + 6·77-s + 8·79-s + 12·83-s + 91-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s − 0.277·13-s − 1.94·17-s − 1.11·29-s + 0.359·31-s − 1.64·37-s + 1.24·41-s − 0.914·43-s − 1.16·47-s + 1/7·49-s − 1.64·53-s − 1.04·59-s + 0.768·61-s + 0.977·67-s + 0.474·71-s − 0.234·73-s + 0.683·77-s + 0.900·79-s + 1.31·83-s + 0.104·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.97975016384074, −12.46694612099801, −12.01628977344293, −11.25859391936338, −10.98366830544244, −10.71344973240231, −10.07282162945516, −9.673780751165912, −9.239862475626571, −8.595710192396344, −8.350978915451476, −7.583250221719327, −7.484954236998671, −6.678800713456092, −6.412480702426639, −5.826891230127392, −5.104046010755216, −4.922700631939509, −4.423689169956808, −3.539117957405177, −3.309280947583810, −2.393621206335115, −2.276099068022802, −1.583704787726629, −0.4567433844173734, 0,
0.4567433844173734, 1.583704787726629, 2.276099068022802, 2.393621206335115, 3.309280947583810, 3.539117957405177, 4.423689169956808, 4.922700631939509, 5.104046010755216, 5.826891230127392, 6.412480702426639, 6.678800713456092, 7.484954236998671, 7.583250221719327, 8.350978915451476, 8.595710192396344, 9.239862475626571, 9.673780751165912, 10.07282162945516, 10.71344973240231, 10.98366830544244, 11.25859391936338, 12.01628977344293, 12.46694612099801, 12.97975016384074
