| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s + 3·13-s + 14-s + 15-s + 16-s − 2·17-s + 18-s − 20-s − 21-s + 9·23-s − 24-s + 25-s + 3·26-s − 27-s + 28-s + 9·29-s + 30-s + 7·31-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 + 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s + 1.87·23-s − 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.182·30-s + 1.25·31-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
| \(L(1)\) |
\(\approx\) |
\(3.711333120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.711333120\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.36008785497301, −14.86247161698594, −14.26847357848699, −13.68684159307865, −13.15810997380616, −12.70603226558765, −12.07845040480493, −11.56876097781609, −11.16395306571461, −10.62230613791833, −10.21389709191914, −9.191130464013425, −8.743474690770435, −8.055849825427975, −7.412292305995513, −6.835648070620310, −6.256258636611941, −5.753587787314878, −4.915510354356531, −4.503461889335524, −4.007336876221451, −2.999481253276337, −2.584968713287886, −1.308867844929669, −0.8031979061492657,
0.8031979061492657, 1.308867844929669, 2.584968713287886, 2.999481253276337, 4.007336876221451, 4.503461889335524, 4.915510354356531, 5.753587787314878, 6.256258636611941, 6.835648070620310, 7.412292305995513, 8.055849825427975, 8.743474690770435, 9.191130464013425, 10.21389709191914, 10.62230613791833, 11.16395306571461, 11.56876097781609, 12.07845040480493, 12.70603226558765, 13.15810997380616, 13.68684159307865, 14.26847357848699, 14.86247161698594, 15.36008785497301
