| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 5·11-s − 13-s + 14-s + 16-s + 17-s − 6·19-s − 20-s − 5·22-s − 6·23-s − 4·25-s − 26-s + 28-s − 6·29-s + 4·31-s + 32-s + 34-s − 35-s + 11·37-s − 6·38-s − 40-s − 9·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.50·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.37·19-s − 0.223·20-s − 1.06·22-s − 1.25·23-s − 4/5·25-s − 0.196·26-s + 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.169·35-s + 1.80·37-s − 0.973·38-s − 0.158·40-s − 1.37·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−8.214818534375034915463026148944, −8.058182215350688246291268937667, −7.17784034658125445256753156098, −6.14279974211800228995769619333, −5.47583971400052312613240501777, −4.56902455627867224907319278116, −3.92372992981631625850996692437, −2.75538936400630092821215712431, −1.94124859436574581607193810396, 0,
1.94124859436574581607193810396, 2.75538936400630092821215712431, 3.92372992981631625850996692437, 4.56902455627867224907319278116, 5.47583971400052312613240501777, 6.14279974211800228995769619333, 7.17784034658125445256753156098, 8.058182215350688246291268937667, 8.214818534375034915463026148944
