| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s − 2·9-s − 3·11-s − 12-s − 4·13-s + 2·14-s + 16-s − 3·17-s + 2·18-s − 5·19-s + 2·21-s + 3·22-s + 6·23-s + 24-s + 4·26-s + 5·27-s − 2·28-s − 2·31-s − 32-s + 3·33-s + 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.14·19-s + 0.436·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 0.784·26-s + 0.962·27-s − 0.377·28-s − 0.359·31-s − 0.176·32-s + 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−14.58608096263009, −13.91947905025749, −13.18757269776805, −12.77960690705417, −12.52264046550732, −11.76246439948684, −11.26819125718530, −10.87150840577458, −10.42536948024190, −9.838553051039779, −9.383597427199642, −8.872705551432373, −8.158458706523908, −7.959560767273526, −6.954141261172424, −6.713035025859479, −6.294149810449032, −5.418121619373889, −5.083250008622917, −4.498151041675857, −3.479136172551121, −2.899389985025451, −2.414951362198195, −1.681708023640057, −0.4819357900718883, 0,
0.4819357900718883, 1.681708023640057, 2.414951362198195, 2.899389985025451, 3.479136172551121, 4.498151041675857, 5.083250008622917, 5.418121619373889, 6.294149810449032, 6.713035025859479, 6.954141261172424, 7.959560767273526, 8.158458706523908, 8.872705551432373, 9.383597427199642, 9.838553051039779, 10.42536948024190, 10.87150840577458, 11.26819125718530, 11.76246439948684, 12.52264046550732, 12.77960690705417, 13.18757269776805, 13.91947905025749, 14.58608096263009
