| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 3·13-s + 16-s + 3·17-s − 6·19-s + 3·20-s + 3·23-s + 4·25-s − 3·26-s − 6·29-s − 4·31-s − 32-s − 3·34-s − 2·37-s + 6·38-s − 3·40-s − 3·41-s + 6·43-s − 3·46-s − 4·50-s + 3·52-s + 9·53-s + 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.832·13-s + 1/4·16-s + 0.727·17-s − 1.37·19-s + 0.670·20-s + 0.625·23-s + 4/5·25-s − 0.588·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.328·37-s + 0.973·38-s − 0.474·40-s − 0.468·41-s + 0.914·43-s − 0.442·46-s − 0.565·50-s + 0.416·52-s + 1.23·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.81416817245561, −12.51656274084200, −11.90828208970505, −11.26162839527358, −10.93773565403465, −10.49627579042408, −10.12707227010195, −9.654501058888552, −9.131741761288819, −8.852366321962065, −8.427896037066578, −7.819751645879354, −7.257455715739430, −6.851347128021962, −6.221398520859714, −5.967005195160227, −5.411912908975333, −5.087484547645661, −4.038502834454442, −3.859280036831787, −2.938707494298845, −2.559481544847659, −1.781102200545084, −1.597993646304449, −0.8490172778279399, 0,
0.8490172778279399, 1.597993646304449, 1.781102200545084, 2.559481544847659, 2.938707494298845, 3.859280036831787, 4.038502834454442, 5.087484547645661, 5.411912908975333, 5.967005195160227, 6.221398520859714, 6.851347128021962, 7.257455715739430, 7.819751645879354, 8.427896037066578, 8.852366321962065, 9.131741761288819, 9.654501058888552, 10.12707227010195, 10.49627579042408, 10.93773565403465, 11.26162839527358, 11.90828208970505, 12.51656274084200, 12.81416817245561
