| L(s) = 1 | + 2·5-s − 2·7-s − 2·13-s + 2·17-s + 6·19-s − 23-s − 25-s + 4·29-s − 8·31-s − 4·35-s − 2·37-s − 12·41-s − 6·43-s − 8·47-s − 3·49-s − 2·53-s + 4·59-s − 4·61-s − 4·65-s + 12·67-s + 12·71-s − 10·73-s + 10·79-s + 4·83-s + 4·85-s − 6·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.554·13-s + 0.485·17-s + 1.37·19-s − 0.208·23-s − 1/5·25-s + 0.742·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s − 1.87·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.520·59-s − 0.512·61-s − 0.496·65-s + 1.46·67-s + 1.42·71-s − 1.17·73-s + 1.12·79-s + 0.439·83-s + 0.433·85-s − 0.635·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.35940216483618, −12.80491891025848, −12.45241172789937, −11.89558734665692, −11.51777923046308, −10.93971344665488, −10.22951695418102, −9.935164359145431, −9.627710532970778, −9.285244293415359, −8.520769226903582, −8.133291634137818, −7.483038867327786, −6.972897503217870, −6.587699053697936, −6.045037260253290, −5.445812667577695, −5.144206264806106, −4.637652344116065, −3.570698884272451, −3.423380642400947, −2.846096482345793, −1.960402869155270, −1.715524137787534, −0.7890028975896674, 0,
0.7890028975896674, 1.715524137787534, 1.960402869155270, 2.846096482345793, 3.423380642400947, 3.570698884272451, 4.637652344116065, 5.144206264806106, 5.445812667577695, 6.045037260253290, 6.587699053697936, 6.972897503217870, 7.483038867327786, 8.133291634137818, 8.520769226903582, 9.285244293415359, 9.627710532970778, 9.935164359145431, 10.22951695418102, 10.93971344665488, 11.51777923046308, 11.89558734665692, 12.45241172789937, 12.80491891025848, 13.35940216483618
