| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 5·11-s − 14-s + 16-s + 17-s − 4·19-s − 20-s + 5·22-s + 23-s − 4·25-s − 28-s + 7·29-s + 10·31-s + 32-s + 34-s + 35-s − 8·37-s − 4·38-s − 40-s − 2·41-s + 43-s + 5·44-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 − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s + 1.06·22-s + 0.208·23-s − 4/5·25-s − 0.188·28-s + 1.29·29-s + 1.79·31-s + 0.176·32-s + 0.171·34-s + 0.169·35-s − 1.31·37-s − 0.648·38-s − 0.158·40-s − 0.312·41-s + 0.152·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.594258907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.594258907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.24736538264089, −12.11670719348205, −11.76109128032280, −11.46565855356784, −10.65712306766202, −10.30534964720365, −9.978669452329834, −9.316193804371189, −8.755944431675785, −8.496166267090298, −7.902979500523655, −7.349209044868486, −6.772227997297140, −6.480111734578230, −6.139559484834872, −5.544605490529782, −4.827461354952180, −4.401852882760231, −4.087624393948566, −3.435302553691321, −3.092342390570591, −2.444400475122480, −1.703388741330781, −1.230420591206439, −0.4392568971511110,
0.4392568971511110, 1.230420591206439, 1.703388741330781, 2.444400475122480, 3.092342390570591, 3.435302553691321, 4.087624393948566, 4.401852882760231, 4.827461354952180, 5.544605490529782, 6.139559484834872, 6.480111734578230, 6.772227997297140, 7.349209044868486, 7.902979500523655, 8.496166267090298, 8.755944431675785, 9.316193804371189, 9.978669452329834, 10.30534964720365, 10.65712306766202, 11.46565855356784, 11.76109128032280, 12.11670719348205, 12.24736538264089
