| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4·11-s + 12-s − 2·13-s + 16-s − 6·17-s − 18-s + 4·22-s + 6·23-s − 24-s + 2·26-s + 27-s + 2·29-s + 6·31-s − 32-s − 4·33-s + 6·34-s + 36-s + 10·37-s − 2·39-s − 6·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.852·22-s + 1.25·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s + 1.64·37-s − 0.320·39-s − 0.914·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.232073971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232073971\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−14.68103877982390, −13.71816232805287, −13.50279802377837, −12.91039141996678, −12.49600717011913, −11.79269742932924, −11.17760747624289, −10.76903325031135, −10.30966100241927, −9.636444416572448, −9.283535846409796, −8.724323636507666, −8.114784641085242, −7.783221784191028, −7.218299550661225, −6.529482304605005, −6.202073503331704, −5.128120851876190, −4.791740209522193, −4.147316537216792, −3.076186710255178, −2.753732108143346, −2.203206418072839, −1.355766673359383, −0.4158752291620293,
0.4158752291620293, 1.355766673359383, 2.203206418072839, 2.753732108143346, 3.076186710255178, 4.147316537216792, 4.791740209522193, 5.128120851876190, 6.202073503331704, 6.529482304605005, 7.218299550661225, 7.783221784191028, 8.114784641085242, 8.724323636507666, 9.283535846409796, 9.636444416572448, 10.30966100241927, 10.76903325031135, 11.17760747624289, 11.79269742932924, 12.49600717011913, 12.91039141996678, 13.50279802377837, 13.71816232805287, 14.68103877982390
