| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 4·11-s − 3·13-s + 15-s − 3·17-s + 7·19-s − 2·21-s + 23-s − 4·25-s − 27-s + 2·29-s + 31-s + 4·33-s − 2·35-s + 8·37-s + 3·39-s + 41-s − 4·43-s − 45-s − 8·47-s − 3·49-s + 3·51-s + 6·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.832·13-s + 0.258·15-s − 0.727·17-s + 1.60·19-s − 0.436·21-s + 0.208·23-s − 4/5·25-s − 0.192·27-s + 0.371·29-s + 0.179·31-s + 0.696·33-s − 0.338·35-s + 1.31·37-s + 0.480·39-s + 0.156·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.420·51-s + 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675872775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675872775\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−13.16058337914078, −12.71737400408747, −12.02842885865102, −11.70593725598740, −11.28239875046177, −11.04988307105522, −10.25049320638592, −9.868934855919557, −9.616640053050097, −8.754802812394819, −8.231198021912411, −7.833401264125390, −7.436772609132094, −6.966481796228240, −6.379201532757819, −5.600257795642865, −5.325068130490982, −4.743155692159867, −4.503175288125547, −3.649565244328150, −3.109748643671765, −2.338430472648904, −1.978750261081056, −0.9507517257535700, −0.4636386052991593,
0.4636386052991593, 0.9507517257535700, 1.978750261081056, 2.338430472648904, 3.109748643671765, 3.649565244328150, 4.503175288125547, 4.743155692159867, 5.325068130490982, 5.600257795642865, 6.379201532757819, 6.966481796228240, 7.436772609132094, 7.833401264125390, 8.231198021912411, 8.754802812394819, 9.616640053050097, 9.868934855919557, 10.25049320638592, 11.04988307105522, 11.28239875046177, 11.70593725598740, 12.02842885865102, 12.71737400408747, 13.16058337914078
