| L(s) = 1 | + 3-s + 3·5-s + 9-s + 11-s − 13-s + 3·15-s − 7·17-s − 19-s + 7·23-s + 4·25-s + 27-s − 3·29-s + 33-s + 5·37-s − 39-s − 4·41-s + 11·43-s + 3·45-s − 7·51-s + 14·53-s + 3·55-s − 57-s − 4·59-s + 61-s − 3·65-s − 6·67-s + 7·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.774·15-s − 1.69·17-s − 0.229·19-s + 1.45·23-s + 4/5·25-s + 0.192·27-s − 0.557·29-s + 0.174·33-s + 0.821·37-s − 0.160·39-s − 0.624·41-s + 1.67·43-s + 0.447·45-s − 0.980·51-s + 1.92·53-s + 0.404·55-s − 0.132·57-s − 0.520·59-s + 0.128·61-s − 0.372·65-s − 0.733·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.513147614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.513147614\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.55412539055238, −13.26358734031856, −12.74094139792661, −12.26501576688383, −11.53883833834460, −11.00151254595424, −10.60327944259806, −10.15975501248822, −9.343788124769082, −9.194038252243868, −8.970853708592899, −8.186937809752277, −7.619852873170118, −6.960025647069870, −6.578794029816161, −6.174278149462491, −5.323427818749470, −5.128796387158580, −4.205155555170546, −3.964868105688341, −2.917845147885567, −2.517659201758212, −2.058804194523377, −1.400181731926767, −0.6096803876404723,
0.6096803876404723, 1.400181731926767, 2.058804194523377, 2.517659201758212, 2.917845147885567, 3.964868105688341, 4.205155555170546, 5.128796387158580, 5.323427818749470, 6.174278149462491, 6.578794029816161, 6.960025647069870, 7.619852873170118, 8.186937809752277, 8.970853708592899, 9.194038252243868, 9.343788124769082, 10.15975501248822, 10.60327944259806, 11.00151254595424, 11.53883833834460, 12.26501576688383, 12.74094139792661, 13.26358734031856, 13.55412539055238
