| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 12-s + 6·13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s + 4·23-s − 24-s + 25-s − 6·26-s + 27-s + 28-s − 4·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.742·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066750387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066750387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−15.32439985748049, −15.05213461948373, −14.29880440194908, −13.79763762107056, −13.15594281589172, −12.80387780766143, −11.88842430077525, −11.50763788537117, −10.88481195128345, −10.57126652123962, −9.792727243358862, −9.114366529321701, −8.723112150277800, −8.250828221914527, −7.761650602862384, −7.001522080701382, −6.637826003731980, −5.782230878183242, −5.111268138099737, −4.307165884465772, −3.417950381553562, −3.287875754382632, −2.056157182577465, −1.524837836038256, −0.6374053479747298,
0.6374053479747298, 1.524837836038256, 2.056157182577465, 3.287875754382632, 3.417950381553562, 4.307165884465772, 5.111268138099737, 5.782230878183242, 6.637826003731980, 7.001522080701382, 7.761650602862384, 8.250828221914527, 8.723112150277800, 9.114366529321701, 9.792727243358862, 10.57126652123962, 10.88481195128345, 11.50763788537117, 11.88842430077525, 12.80387780766143, 13.15594281589172, 13.79763762107056, 14.29880440194908, 15.05213461948373, 15.32439985748049
