| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s + 13-s + 16-s − 5·17-s + 18-s + 7·19-s + 4·22-s + 8·23-s + 24-s + 26-s + 27-s + 2·29-s − 2·31-s + 32-s + 4·33-s − 5·34-s + 36-s − 6·37-s + 7·38-s + 39-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 + 1.20·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s − 1.21·17-s + 0.235·18-s + 1.60·19-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.176·32-s + 0.696·33-s − 0.857·34-s + 1/6·36-s − 0.986·37-s + 1.13·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.232712098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.232712098\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−13.78082222641610, −13.34503595805556, −13.06696618893213, −12.31832002547594, −11.89311676070717, −11.39919490397887, −11.07774894005817, −10.31925903344786, −9.882796064408092, −9.177803565971495, −8.817652657973198, −8.495444427267291, −7.584017392045970, −7.031494905889885, −6.851572360155213, −6.249162719015180, −5.333487998968687, −5.170974757448183, −4.309133228954781, −3.894598365992031, −3.288827366195551, −2.841832087895961, −2.051770948334541, −1.396281124533260, −0.7625129754365079,
0.7625129754365079, 1.396281124533260, 2.051770948334541, 2.841832087895961, 3.288827366195551, 3.894598365992031, 4.309133228954781, 5.170974757448183, 5.333487998968687, 6.249162719015180, 6.851572360155213, 7.031494905889885, 7.584017392045970, 8.495444427267291, 8.817652657973198, 9.177803565971495, 9.882796064408092, 10.31925903344786, 11.07774894005817, 11.39919490397887, 11.89311676070717, 12.31832002547594, 13.06696618893213, 13.34503595805556, 13.78082222641610
