| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 3·11-s + 16-s − 7·17-s − 3·19-s − 20-s − 3·22-s + 23-s − 4·25-s + 29-s + 8·31-s − 32-s + 7·34-s + 37-s + 3·38-s + 40-s + 4·41-s + 5·43-s + 3·44-s − 46-s + 4·50-s + 6·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.904·11-s + 1/4·16-s − 1.69·17-s − 0.688·19-s − 0.223·20-s − 0.639·22-s + 0.208·23-s − 4/5·25-s + 0.185·29-s + 1.43·31-s − 0.176·32-s + 1.20·34-s + 0.164·37-s + 0.486·38-s + 0.158·40-s + 0.624·41-s + 0.762·43-s + 0.452·44-s − 0.147·46-s + 0.565·50-s + 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8349105452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8349105452\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.29448562210163, −12.89117620852105, −12.15136230666603, −11.91466096771525, −11.27118492303100, −11.10667047324184, −10.44570558823860, −9.969104255160968, −9.448451788569320, −8.891727174582575, −8.588205080598928, −8.176173024657007, −7.394766203371561, −7.135018738477950, −6.430963517032028, −6.199722111544649, −5.550985244591632, −4.613463196052517, −4.240515767172186, −3.900919721906427, −2.892037053614419, −2.532641219106085, −1.778075340448296, −1.156548107108500, −0.3221967214394491,
0.3221967214394491, 1.156548107108500, 1.778075340448296, 2.532641219106085, 2.892037053614419, 3.900919721906427, 4.240515767172186, 4.613463196052517, 5.550985244591632, 6.199722111544649, 6.430963517032028, 7.135018738477950, 7.394766203371561, 8.176173024657007, 8.588205080598928, 8.891727174582575, 9.448451788569320, 9.969104255160968, 10.44570558823860, 11.10667047324184, 11.27118492303100, 11.91466096771525, 12.15136230666603, 12.89117620852105, 13.29448562210163
