| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s − 13-s + 14-s + 16-s − 5·17-s + 4·19-s + 2·20-s + 7·23-s − 25-s − 26-s + 28-s − 5·29-s − 10·31-s + 32-s − 5·34-s + 2·35-s − 4·37-s + 4·38-s + 2·40-s − 8·41-s − 43-s + 7·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s + 0.917·19-s + 0.447·20-s + 1.45·23-s − 1/5·25-s − 0.196·26-s + 0.188·28-s − 0.928·29-s − 1.79·31-s + 0.176·32-s − 0.857·34-s + 0.338·35-s − 0.657·37-s + 0.648·38-s + 0.316·40-s − 1.24·41-s − 0.152·43-s + 1.03·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.34655444698130, −12.95311271852645, −12.56556299903079, −11.77862516607449, −11.55416137289518, −10.99603737266655, −10.58948186142175, −10.13805298326927, −9.351076354553198, −9.221983805202588, −8.676443073371563, −7.983566075340725, −7.354855266039658, −6.979666918645879, −6.641570799774915, −5.789970590610700, −5.519621322990323, −5.066209914275188, −4.619091473922376, −3.835812937407295, −3.433570064097722, −2.747666087731939, −2.044415283632893, −1.820021443407424, −1.006220037432616, 0,
1.006220037432616, 1.820021443407424, 2.044415283632893, 2.747666087731939, 3.433570064097722, 3.835812937407295, 4.619091473922376, 5.066209914275188, 5.519621322990323, 5.789970590610700, 6.641570799774915, 6.979666918645879, 7.354855266039658, 7.983566075340725, 8.676443073371563, 9.221983805202588, 9.351076354553198, 10.13805298326927, 10.58948186142175, 10.99603737266655, 11.55416137289518, 11.77862516607449, 12.56556299903079, 12.95311271852645, 13.34655444698130
