| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s + 11-s − 12-s + 4·13-s − 2·14-s + 16-s + 17-s + 18-s + 8·19-s + 2·21-s + 22-s − 3·23-s − 24-s + 4·26-s − 27-s − 2·28-s + 6·29-s + 8·31-s + 32-s − 33-s + 34-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 + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.436·21-s + 0.213·22-s − 0.625·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.681060728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.681060728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.36708216994355, −14.55103000944538, −14.07782924670673, −13.61130806721594, −13.05646437728576, −12.67500713203131, −11.86178055332795, −11.60204579562605, −11.23068069949753, −10.20684256450785, −10.01809711408121, −9.450040462007265, −8.539266521346821, −8.041162965484816, −7.270302189204677, −6.702654348533014, −6.119134737409904, −5.832666087892625, −5.016909149193096, −4.451108596372350, −3.668159786759977, −3.213325340233401, −2.475291020996545, −1.307244531477160, −0.7778972001944370,
0.7778972001944370, 1.307244531477160, 2.475291020996545, 3.213325340233401, 3.668159786759977, 4.451108596372350, 5.016909149193096, 5.832666087892625, 6.119134737409904, 6.702654348533014, 7.270302189204677, 8.041162965484816, 8.539266521346821, 9.450040462007265, 10.01809711408121, 10.20684256450785, 11.23068069949753, 11.60204579562605, 11.86178055332795, 12.67500713203131, 13.05646437728576, 13.61130806721594, 14.07782924670673, 14.55103000944538, 15.36708216994355
