| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 2·11-s − 4·14-s + 16-s + 4·17-s − 2·19-s − 2·22-s − 6·23-s + 4·28-s + 2·29-s + 4·31-s − 32-s − 4·34-s − 6·37-s + 2·38-s − 6·41-s − 8·43-s + 2·44-s + 6·46-s − 8·47-s + 9·49-s + 10·53-s − 4·56-s − 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 0.603·11-s − 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s − 0.426·22-s − 1.25·23-s + 0.755·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.986·37-s + 0.324·38-s − 0.937·41-s − 1.21·43-s + 0.301·44-s + 0.884·46-s − 1.16·47-s + 9/7·49-s + 1.37·53-s − 0.534·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.31451400006862, −13.98533366717733, −13.43850515250457, −12.59838211159448, −12.06382483844837, −11.78027438178623, −11.34425432702484, −10.77002960711953, −10.14376703277425, −9.936096015702638, −9.224378807830812, −8.482376579008750, −8.186243957026958, −8.012717575522153, −7.089142951934285, −6.765323365144889, −6.056481376868689, −5.406642743036708, −4.955456292010916, −4.245451484746157, −3.680615787978798, −2.935406941411924, −2.052084376405728, −1.625194993356706, −1.046853712280038, 0,
1.046853712280038, 1.625194993356706, 2.052084376405728, 2.935406941411924, 3.680615787978798, 4.245451484746157, 4.955456292010916, 5.406642743036708, 6.056481376868689, 6.765323365144889, 7.089142951934285, 8.012717575522153, 8.186243957026958, 8.482376579008750, 9.224378807830812, 9.936096015702638, 10.14376703277425, 10.77002960711953, 11.34425432702484, 11.78027438178623, 12.06382483844837, 12.59838211159448, 13.43850515250457, 13.98533366717733, 14.31451400006862
