| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 5·11-s + 13-s + 16-s − 3·17-s + 19-s − 20-s + 5·22-s − 3·23-s − 4·25-s − 26-s − 9·29-s − 4·31-s − 32-s + 3·34-s − 11·37-s − 38-s + 40-s − 5·43-s − 5·44-s + 3·46-s − 8·47-s + 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.50·11-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.229·19-s − 0.223·20-s + 1.06·22-s − 0.625·23-s − 4/5·25-s − 0.196·26-s − 1.67·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s − 1.80·37-s − 0.162·38-s + 0.158·40-s − 0.762·43-s − 0.753·44-s + 0.442·46-s − 1.16·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3331336457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3331336457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 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
−16.20263860351110, −16.02760519824951, −15.46034320855662, −14.96875520717892, −14.24836431155714, −13.38923686616624, −13.09430702159273, −12.40601697474404, −11.56716584634695, −11.29512068430796, −10.56217948200760, −10.08187724139811, −9.469308811930295, −8.645432877164782, −8.256414417694115, −7.553190592959670, −7.156964974796266, −6.311173935233802, −5.483709014651115, −5.044464728232113, −3.889354199365660, −3.386069913994935, −2.304436416253823, −1.765032339268031, −0.2795961366953430,
0.2795961366953430, 1.765032339268031, 2.304436416253823, 3.386069913994935, 3.889354199365660, 5.044464728232113, 5.483709014651115, 6.311173935233802, 7.156964974796266, 7.553190592959670, 8.256414417694115, 8.645432877164782, 9.469308811930295, 10.08187724139811, 10.56217948200760, 11.29512068430796, 11.56716584634695, 12.40601697474404, 13.09430702159273, 13.38923686616624, 14.24836431155714, 14.96875520717892, 15.46034320855662, 16.02760519824951, 16.20263860351110
