| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s − 4·7-s + 3·8-s + 9-s − 12-s + 4·14-s − 16-s − 6·17-s − 18-s + 4·19-s − 4·21-s − 23-s + 3·24-s − 5·25-s + 27-s + 4·28-s − 4·29-s + 4·31-s − 5·32-s + 6·34-s − 36-s + 2·37-s − 4·38-s + 12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.872·21-s − 0.208·23-s + 0.612·24-s − 25-s + 0.192·27-s + 0.755·28-s − 0.742·29-s + 0.718·31-s − 0.883·32-s + 1.02·34-s − 1/6·36-s + 0.328·37-s − 0.648·38-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152421 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| 47 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.58331539301582, −13.20256199615950, −12.76618983239928, −12.25625800966985, −11.58693324932475, −11.06859777164519, −10.50759066599796, −9.988946327842864, −9.535120792045461, −9.336945568936671, −8.965702479451028, −8.181542985288638, −7.952192116600196, −7.293234777057295, −6.829677746589899, −6.295708425915447, −5.780204738826764, −5.022480380129099, −4.455570509106062, −3.879619645753943, −3.478443544380128, −2.784592707795909, −2.197294501269120, −1.490504056220149, −0.6190401062613273, 0,
0.6190401062613273, 1.490504056220149, 2.197294501269120, 2.784592707795909, 3.478443544380128, 3.879619645753943, 4.455570509106062, 5.022480380129099, 5.780204738826764, 6.295708425915447, 6.829677746589899, 7.293234777057295, 7.952192116600196, 8.181542985288638, 8.965702479451028, 9.336945568936671, 9.535120792045461, 9.988946327842864, 10.50759066599796, 11.06859777164519, 11.58693324932475, 12.25625800966985, 12.76618983239928, 13.20256199615950, 13.58331539301582
