| L(s) = 1 | − 4·5-s + 2·11-s − 2·13-s + 2·17-s − 2·19-s + 23-s + 11·25-s + 6·29-s + 4·37-s − 10·41-s + 10·43-s + 8·47-s − 8·55-s + 4·61-s + 8·65-s − 2·67-s + 8·71-s + 2·73-s + 8·79-s + 6·83-s − 8·85-s + 6·89-s + 8·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.458·19-s + 0.208·23-s + 11/5·25-s + 1.11·29-s + 0.657·37-s − 1.56·41-s + 1.52·43-s + 1.16·47-s − 1.07·55-s + 0.512·61-s + 0.992·65-s − 0.244·67-s + 0.949·71-s + 0.234·73-s + 0.900·79-s + 0.658·83-s − 0.867·85-s + 0.635·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.51597567800066, −12.85543188676658, −12.32950379627651, −12.06002300126397, −11.80744683963860, −11.11466914126041, −10.78338462545411, −10.29238055670845, −9.642680019374952, −9.067677537753942, −8.678157328452463, −7.992219605448002, −7.860002030639425, −7.242043813930900, −6.696961145778875, −6.405318867924262, −5.469659135168385, −4.993335470235922, −4.414981994476572, −3.929577382909732, −3.610889628827016, −2.821221750073094, −2.403814407503417, −1.293008856700023, −0.7623321430658803, 0,
0.7623321430658803, 1.293008856700023, 2.403814407503417, 2.821221750073094, 3.610889628827016, 3.929577382909732, 4.414981994476572, 4.993335470235922, 5.469659135168385, 6.405318867924262, 6.696961145778875, 7.242043813930900, 7.860002030639425, 7.992219605448002, 8.678157328452463, 9.067677537753942, 9.642680019374952, 10.29238055670845, 10.78338462545411, 11.11466914126041, 11.80744683963860, 12.06002300126397, 12.32950379627651, 12.85543188676658, 13.51597567800066
