| L(s) = 1 | + 2.56·2-s + 3·3-s − 1.43·4-s − 5·5-s + 7.68·6-s − 6.24·7-s − 24.1·8-s + 9·9-s − 12.8·10-s − 4.31·12-s + 49.1·13-s − 16·14-s − 15·15-s − 50.4·16-s − 82.7·17-s + 23.0·18-s + 130.·19-s + 7.19·20-s − 18.7·21-s − 185.·23-s − 72.5·24-s + 25·25-s + 125.·26-s + 27·27-s + 8.98·28-s + 8.90·29-s − 38.4·30-s + ⋯ |
| L(s) = 1 | + 0.905·2-s + 0.577·3-s − 0.179·4-s − 0.447·5-s + 0.522·6-s − 0.337·7-s − 1.06·8-s + 0.333·9-s − 0.405·10-s − 0.103·12-s + 1.04·13-s − 0.305·14-s − 0.258·15-s − 0.787·16-s − 1.17·17-s + 0.301·18-s + 1.57·19-s + 0.0804·20-s − 0.194·21-s − 1.68·23-s − 0.616·24-s + 0.200·25-s + 0.949·26-s + 0.192·27-s + 0.0606·28-s + 0.0570·29-s − 0.233·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.913040835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913040835\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.56T + 8T^{2} \) |
| 7 | \( 1 + 6.24T + 343T^{2} \) |
| 13 | \( 1 - 49.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 5.26T + 2.97e4T^{2} \) |
| 37 | \( 1 + 416.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 513.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 557.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 168.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 786.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 339.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 309.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 798.T + 9.12e5T^{2} \) |
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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
−8.975919455955815981495464874013, −8.152947976751197832704830214692, −7.32152660429628208049124838681, −6.31930385897685270299856268448, −5.64792002313435453755987444478, −4.58659632820078598924933687549, −3.84712494123525724871669034581, −3.30519085923496854215335388401, −2.19846725010747700947176833085, −0.66931489168708182584882321953,
0.66931489168708182584882321953, 2.19846725010747700947176833085, 3.30519085923496854215335388401, 3.84712494123525724871669034581, 4.58659632820078598924933687549, 5.64792002313435453755987444478, 6.31930385897685270299856268448, 7.32152660429628208049124838681, 8.152947976751197832704830214692, 8.975919455955815981495464874013
