| L(s) = 1 | + 5·7-s − 7·13-s + 8·19-s + 11·31-s − 10·37-s + 8·43-s + 18·49-s − 61-s − 16·67-s + 17·73-s + 17·79-s − 35·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 1.94·13-s + 1.83·19-s + 1.97·31-s − 1.64·37-s + 1.21·43-s + 18/7·49-s − 0.128·61-s − 1.95·67-s + 1.98·73-s + 1.91·79-s − 3.66·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.992447285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.992447285\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.39471389884803, −14.77997512703687, −14.33693422868868, −13.85381936892486, −13.58410513777187, −12.29359799887576, −12.21307906850482, −11.69712165598981, −11.17325925429447, −10.41469275205790, −10.05258646001768, −9.300560809656607, −8.811076170011702, −7.897934944463398, −7.732173362543247, −7.225022450875451, −6.430747438204517, −5.422782219406493, −5.034775489303719, −4.728592450605792, −3.893809216927398, −2.881521713032035, −2.317189610752819, −1.509066707071880, −0.7226878591975072,
0.7226878591975072, 1.509066707071880, 2.317189610752819, 2.881521713032035, 3.893809216927398, 4.728592450605792, 5.034775489303719, 5.422782219406493, 6.430747438204517, 7.225022450875451, 7.732173362543247, 7.897934944463398, 8.811076170011702, 9.300560809656607, 10.05258646001768, 10.41469275205790, 11.17325925429447, 11.69712165598981, 12.21307906850482, 12.29359799887576, 13.58410513777187, 13.85381936892486, 14.33693422868868, 14.77997512703687, 15.39471389884803
