| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s + 4·7-s − 3·8-s + 9-s + 10-s + 12-s − 4·13-s + 4·14-s − 15-s − 16-s + 4·17-s + 18-s − 20-s − 4·21-s + 3·24-s + 25-s − 4·26-s − 27-s − 4·28-s − 8·29-s − 30-s − 8·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.10·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.223·20-s − 0.872·21-s + 0.612·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.755·28-s − 1.48·29-s − 0.182·30-s − 1.43·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.244518338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244518338\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−15.20437131716386, −14.51512861237566, −14.35356247144471, −13.98187432975068, −13.11250493384475, −12.67322766872350, −12.17884479603747, −11.77895363134946, −10.94001978755448, −10.75868794638046, −9.834891756457829, −9.333086509415785, −8.911314816335085, −7.923282826568005, −7.638381682716216, −6.952565343901042, −5.969295828816213, −5.517720945659879, −5.141063090304714, −4.626742168166332, −3.957822953791911, −3.236916572489882, −2.206545986597652, −1.608985751881379, −0.5534639396014143,
0.5534639396014143, 1.608985751881379, 2.206545986597652, 3.236916572489882, 3.957822953791911, 4.626742168166332, 5.141063090304714, 5.517720945659879, 5.969295828816213, 6.952565343901042, 7.638381682716216, 7.923282826568005, 8.911314816335085, 9.333086509415785, 9.834891756457829, 10.75868794638046, 10.94001978755448, 11.77895363134946, 12.17884479603747, 12.67322766872350, 13.11250493384475, 13.98187432975068, 14.35356247144471, 14.51512861237566, 15.20437131716386
