| L(s) = 1 | − 3-s + 9-s + 6·17-s − 4·23-s − 27-s − 10·29-s − 6·37-s − 2·41-s − 4·43-s − 7·49-s − 6·51-s + 6·53-s + 6·61-s − 4·67-s + 4·69-s + 16·71-s − 2·73-s + 81-s − 4·83-s + 10·87-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 6·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.45·17-s − 0.834·23-s − 0.192·27-s − 1.85·29-s − 0.986·37-s − 0.312·41-s − 0.609·43-s − 49-s − 0.840·51-s + 0.824·53-s + 0.768·61-s − 0.488·67-s + 0.481·69-s + 1.89·71-s − 0.234·73-s + 1/9·81-s − 0.439·83-s + 1.07·87-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.237839188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.237839188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.92280555623341, −12.64019514216595, −11.96895624677645, −11.71170202542030, −11.30391647903335, −10.64921546085594, −10.21915853269550, −9.890542396641557, −9.352555021927526, −8.843940029057325, −8.187091893722113, −7.767205299561588, −7.340319075807313, −6.783453045956309, −6.226122531118742, −5.735700786449318, −5.267382225234546, −4.932250744615634, −4.096980643445969, −3.585571407236725, −3.284314006113681, −2.282604011718207, −1.797643749017864, −1.144194041297169, −0.3453047296539916,
0.3453047296539916, 1.144194041297169, 1.797643749017864, 2.282604011718207, 3.284314006113681, 3.585571407236725, 4.096980643445969, 4.932250744615634, 5.267382225234546, 5.735700786449318, 6.226122531118742, 6.783453045956309, 7.340319075807313, 7.767205299561588, 8.187091893722113, 8.843940029057325, 9.352555021927526, 9.890542396641557, 10.21915853269550, 10.64921546085594, 11.30391647903335, 11.71170202542030, 11.96895624677645, 12.64019514216595, 12.92280555623341
