| L(s) = 1 | − 3-s − 4·5-s + 7-s + 9-s + 2·11-s + 2·13-s + 4·15-s − 4·19-s − 21-s − 6·23-s + 11·25-s − 27-s − 10·29-s − 8·31-s − 2·33-s − 4·35-s + 10·37-s − 2·39-s + 4·41-s − 8·43-s − 4·45-s + 4·47-s + 49-s − 10·53-s − 8·55-s + 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.917·19-s − 0.218·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s + 0.624·41-s − 1.21·43-s − 0.596·45-s + 0.583·47-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8010425187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8010425187\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−12.43734851866825, −11.81909146909128, −11.55613925832805, −11.09629738284852, −10.99005203137282, −10.43177851268513, −9.642425371906928, −9.349173110124695, −8.698498513646919, −8.235650747446820, −7.954695609659734, −7.333178244964659, −7.164927689369755, −6.406147660942755, −6.033092385518887, −5.509584221271152, −4.867599908256428, −4.215951190558519, −4.122990603955860, −3.594931423163173, −3.130952544420166, −2.081387295996789, −1.750876668860316, −0.8533763624282939, −0.3057632552232025,
0.3057632552232025, 0.8533763624282939, 1.750876668860316, 2.081387295996789, 3.130952544420166, 3.594931423163173, 4.122990603955860, 4.215951190558519, 4.867599908256428, 5.509584221271152, 6.033092385518887, 6.406147660942755, 7.164927689369755, 7.333178244964659, 7.954695609659734, 8.235650747446820, 8.698498513646919, 9.349173110124695, 9.642425371906928, 10.43177851268513, 10.99005203137282, 11.09629738284852, 11.55613925832805, 11.81909146909128, 12.43734851866825
