| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s + 11-s − 12-s + 2·13-s + 14-s − 2·15-s + 16-s + 6·17-s − 18-s − 8·19-s + 2·20-s + 21-s − 22-s + 4·23-s + 24-s − 25-s − 2·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.218·21-s − 0.213·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001860026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001860026\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−10.75337831344632677723423902927, −10.20625320023432466138451306866, −9.376546318447934246018116649583, −8.498418002269993531963236271910, −7.34608818311813371377464448078, −6.21868608096758870562967590194, −5.83815977597462037088554178569, −4.27546340147020629083452713682, −2.65895736335338182549695498478, −1.14483701940607571217981775775,
1.14483701940607571217981775775, 2.65895736335338182549695498478, 4.27546340147020629083452713682, 5.83815977597462037088554178569, 6.21868608096758870562967590194, 7.34608818311813371377464448078, 8.498418002269993531963236271910, 9.376546318447934246018116649583, 10.20625320023432466138451306866, 10.75337831344632677723423902927
