| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 2·13-s − 15-s + 6·17-s − 2·21-s + 25-s + 27-s − 4·29-s + 8·31-s + 2·35-s + 6·37-s + 2·39-s + 6·43-s − 45-s + 8·47-s − 3·49-s + 6·51-s + 14·53-s − 4·59-s + 8·61-s − 2·63-s − 2·65-s + 12·67-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.436·21-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.320·39-s + 0.914·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.840·51-s + 1.92·53-s − 0.520·59-s + 1.02·61-s − 0.251·63-s − 0.248·65-s + 1.46·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.772955000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.772955000\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.68959063982920, −13.09190875513895, −12.52051990655023, −12.42727737901211, −11.53193869249132, −11.37698872534464, −10.51351761963905, −10.15670792507141, −9.680890448916566, −9.200298814733204, −8.681116705445842, −8.081242566871894, −7.754399068900175, −7.220749444237295, −6.607099522207275, −6.109781391855971, −5.548841463432681, −4.959117128787321, −4.141532725938393, −3.745987728221263, −3.307753482430600, −2.632776492574125, −2.120179584504214, −0.9828060217808849, −0.7262672373247210,
0.7262672373247210, 0.9828060217808849, 2.120179584504214, 2.632776492574125, 3.307753482430600, 3.745987728221263, 4.141532725938393, 4.959117128787321, 5.548841463432681, 6.109781391855971, 6.607099522207275, 7.220749444237295, 7.754399068900175, 8.081242566871894, 8.681116705445842, 9.200298814733204, 9.680890448916566, 10.15670792507141, 10.51351761963905, 11.37698872534464, 11.53193869249132, 12.42727737901211, 12.52051990655023, 13.09190875513895, 13.68959063982920
