| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 11-s + 16-s − 17-s + 19-s + 20-s + 22-s + 3·23-s − 4·25-s − 9·29-s + 4·31-s + 32-s − 34-s + 9·37-s + 38-s + 40-s + 8·41-s − 7·43-s + 44-s + 3·46-s − 8·47-s − 4·50-s + 10·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 0.223·20-s + 0.213·22-s + 0.625·23-s − 4/5·25-s − 1.67·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s + 1.47·37-s + 0.162·38-s + 0.158·40-s + 1.24·41-s − 1.06·43-s + 0.150·44-s + 0.442·46-s − 1.16·47-s − 0.565·50-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.51280641822304, −13.04975116965316, −12.94200527408597, −12.12401528406069, −11.67885581421168, −11.31804789412521, −10.89515887657558, −10.15852489442802, −9.874565298711069, −9.171833852110127, −8.978908133630349, −8.041376748770744, −7.751772391595160, −7.126665478254369, −6.637265956731092, −6.047942218372746, −5.705970973283917, −5.177249056275239, −4.476802169461088, −4.123133714707382, −3.454513212515387, −2.865170146639522, −2.294488885438494, −1.655733390085369, −1.038017159443796, 0,
1.038017159443796, 1.655733390085369, 2.294488885438494, 2.865170146639522, 3.454513212515387, 4.123133714707382, 4.476802169461088, 5.177249056275239, 5.705970973283917, 6.047942218372746, 6.637265956731092, 7.126665478254369, 7.751772391595160, 8.041376748770744, 8.978908133630349, 9.171833852110127, 9.874565298711069, 10.15852489442802, 10.89515887657558, 11.31804789412521, 11.67885581421168, 12.12401528406069, 12.94200527408597, 13.04975116965316, 13.51280641822304
