| L(s) = 1 | + 0.707·2-s + 0.0372·3-s − 1.49·4-s + 5-s + 0.0263·6-s − 3.49·7-s − 2.47·8-s − 2.99·9-s + 0.707·10-s − 0.0558·12-s + 13-s − 2.47·14-s + 0.0372·15-s + 1.24·16-s + 8.22·17-s − 2.12·18-s + 1.20·19-s − 1.49·20-s − 0.130·21-s − 6.68·23-s − 0.0921·24-s + 25-s + 0.707·26-s − 0.223·27-s + 5.24·28-s + 9.21·29-s + 0.0263·30-s + ⋯ |
| L(s) = 1 | + 0.500·2-s + 0.0214·3-s − 0.749·4-s + 0.447·5-s + 0.0107·6-s − 1.32·7-s − 0.875·8-s − 0.999·9-s + 0.223·10-s − 0.0161·12-s + 0.277·13-s − 0.660·14-s + 0.00960·15-s + 0.312·16-s + 1.99·17-s − 0.499·18-s + 0.276·19-s − 0.335·20-s − 0.0283·21-s − 1.39·23-s − 0.0188·24-s + 0.200·25-s + 0.138·26-s − 0.0429·27-s + 0.990·28-s + 1.71·29-s + 0.00480·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.707T + 2T^{2} \) |
| 3 | \( 1 - 0.0372T + 3T^{2} \) |
| 7 | \( 1 + 3.49T + 7T^{2} \) |
| 17 | \( 1 - 8.22T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 - 0.854T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 + 5.07T + 59T^{2} \) |
| 61 | \( 1 - 1.20T + 61T^{2} \) |
| 67 | \( 1 - 9.90T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 8.05T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.55283432770098649956217243843, −6.35784373905585916983125515952, −6.14345117066236350166999452714, −5.44876930299268758342390014344, −4.80390531259643108121822274813, −3.67817281179155868106855886476, −3.27209441434405259978445281544, −2.62411575564931888533334654592, −1.10095289640126905648851618977, 0,
1.10095289640126905648851618977, 2.62411575564931888533334654592, 3.27209441434405259978445281544, 3.67817281179155868106855886476, 4.80390531259643108121822274813, 5.44876930299268758342390014344, 6.14345117066236350166999452714, 6.35784373905585916983125515952, 7.55283432770098649956217243843
