| L(s) = 1 | + 2·5-s + 7-s + 6·13-s − 2·19-s − 25-s + 8·29-s + 2·35-s − 2·37-s + 2·41-s + 8·43-s + 8·47-s + 49-s + 2·53-s − 12·59-s + 4·61-s + 12·65-s + 12·67-s − 8·73-s + 8·79-s − 10·89-s + 6·91-s − 4·95-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 1.66·13-s − 0.458·19-s − 1/5·25-s + 1.48·29-s + 0.338·35-s − 0.328·37-s + 0.312·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s + 0.512·61-s + 1.48·65-s + 1.46·67-s − 0.936·73-s + 0.900·79-s − 1.05·89-s + 0.628·91-s − 0.410·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.081977801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.081977801\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.97430800612745, −13.75219307620529, −13.18557983985612, −12.64043199326678, −12.16710739930725, −11.54475863572892, −10.94115086400176, −10.64136379185375, −10.16707547304933, −9.450709140111622, −9.068422916325807, −8.446361516442434, −8.148034237227246, −7.412138415575014, −6.686632604404810, −6.296297486186707, −5.721817018846233, −5.392498879832578, −4.442466329813545, −4.122329016023037, −3.330511958101592, −2.637354688048276, −1.996518212710818, −1.328893009972771, −0.7125986581304658,
0.7125986581304658, 1.328893009972771, 1.996518212710818, 2.637354688048276, 3.330511958101592, 4.122329016023037, 4.442466329813545, 5.392498879832578, 5.721817018846233, 6.296297486186707, 6.686632604404810, 7.412138415575014, 8.148034237227246, 8.446361516442434, 9.068422916325807, 9.450709140111622, 10.16707547304933, 10.64136379185375, 10.94115086400176, 11.54475863572892, 12.16710739930725, 12.64043199326678, 13.18557983985612, 13.75219307620529, 13.97430800612745
