| L(s) = 1 | − 3·9-s + 6·11-s + 4·13-s + 7·17-s + 4·19-s + 23-s + 6·29-s − 3·31-s − 4·37-s − 9·41-s + 8·43-s − 11·47-s − 4·53-s + 10·59-s − 2·61-s + 8·67-s − 3·71-s + 2·73-s − 17·79-s + 9·81-s + 2·83-s − 7·89-s − 97-s − 18·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s + 1.80·11-s + 1.10·13-s + 1.69·17-s + 0.917·19-s + 0.208·23-s + 1.11·29-s − 0.538·31-s − 0.657·37-s − 1.40·41-s + 1.21·43-s − 1.60·47-s − 0.549·53-s + 1.30·59-s − 0.256·61-s + 0.977·67-s − 0.356·71-s + 0.234·73-s − 1.91·79-s + 81-s + 0.219·83-s − 0.741·89-s − 0.101·97-s − 1.80·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.516754625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.516754625\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−14.01273377545151, −13.79279846921650, −13.01080358187037, −12.39921864433202, −11.93093662675764, −11.54415629469872, −11.25294461525684, −10.50679443999762, −9.918585479666589, −9.496056978998235, −8.911102919790627, −8.438795987668519, −8.109509523787328, −7.267874562364245, −6.796362792577124, −6.218230481768580, −5.740787891103083, −5.276318676410443, −4.542763605011333, −3.727153954347551, −3.374015454423693, −2.974713274596224, −1.816758779656928, −1.255596077141062, −0.6884952226076359,
0.6884952226076359, 1.255596077141062, 1.816758779656928, 2.974713274596224, 3.374015454423693, 3.727153954347551, 4.542763605011333, 5.276318676410443, 5.740787891103083, 6.218230481768580, 6.796362792577124, 7.267874562364245, 8.109509523787328, 8.438795987668519, 8.911102919790627, 9.496056978998235, 9.918585479666589, 10.50679443999762, 11.25294461525684, 11.54415629469872, 11.93093662675764, 12.39921864433202, 13.01080358187037, 13.79279846921650, 14.01273377545151
