| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 11-s − 12-s + 2·13-s + 16-s + 17-s + 18-s + 22-s − 8·23-s − 24-s + 2·26-s − 27-s − 6·29-s − 8·31-s + 32-s − 33-s + 34-s + 36-s + 10·37-s − 2·39-s − 6·41-s − 12·43-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 + 0.301·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.213·22-s − 1.66·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.518589846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.518589846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.10428819799713, −14.64932007531850, −14.27651123356816, −13.38005835249227, −13.21065526422902, −12.60790416170558, −11.96364688331979, −11.43686664453858, −11.24360356337301, −10.43012046195058, −9.871324075541606, −9.450261995848064, −8.501047331015885, −8.012420093902738, −7.378015113230341, −6.641786718945488, −6.261476416846881, −5.589196813624408, −5.157160615915861, −4.375916391705051, −3.690490636514762, −3.372656051245718, −2.094685637889798, −1.714375282622581, −0.5550085925707754,
0.5550085925707754, 1.714375282622581, 2.094685637889798, 3.372656051245718, 3.690490636514762, 4.375916391705051, 5.157160615915861, 5.589196813624408, 6.261476416846881, 6.641786718945488, 7.378015113230341, 8.012420093902738, 8.501047331015885, 9.450261995848064, 9.871324075541606, 10.43012046195058, 11.24360356337301, 11.43686664453858, 11.96364688331979, 12.60790416170558, 13.21065526422902, 13.38005835249227, 14.27651123356816, 14.64932007531850, 15.10428819799713
