| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 5·13-s − 4·14-s + 16-s + 2·19-s + 3·23-s + 5·26-s + 4·28-s + 6·29-s − 4·31-s − 32-s + 37-s − 2·38-s − 8·43-s − 3·46-s + 3·47-s + 9·49-s − 5·52-s − 6·53-s − 4·56-s − 6·58-s + 9·59-s − 61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1.38·13-s − 1.06·14-s + 1/4·16-s + 0.458·19-s + 0.625·23-s + 0.980·26-s + 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.164·37-s − 0.324·38-s − 1.21·43-s − 0.442·46-s + 0.437·47-s + 9/7·49-s − 0.693·52-s − 0.824·53-s − 0.534·56-s − 0.787·58-s + 1.17·59-s − 0.128·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.97639733661181, −13.18097447966962, −12.64143190140126, −12.11338086541330, −11.72301221708205, −11.31802579979940, −10.80313052652886, −10.38212047417320, −9.745311894945319, −9.457389909037315, −8.768137755820095, −8.253145046389293, −7.992301005037203, −7.336826590434377, −7.020790447715641, −6.461144659410996, −5.561289214162716, −5.183423908570794, −4.759386676083924, −4.170270490207187, −3.339617244971811, −2.650788207785849, −2.164319761758147, −1.494632529793773, −0.9223779827643340, 0,
0.9223779827643340, 1.494632529793773, 2.164319761758147, 2.650788207785849, 3.339617244971811, 4.170270490207187, 4.759386676083924, 5.183423908570794, 5.561289214162716, 6.461144659410996, 7.020790447715641, 7.336826590434377, 7.992301005037203, 8.253145046389293, 8.768137755820095, 9.457389909037315, 9.745311894945319, 10.38212047417320, 10.80313052652886, 11.31802579979940, 11.72301221708205, 12.11338086541330, 12.64143190140126, 13.18097447966962, 13.97639733661181
