| L(s) = 1 | − 1.56·5-s + 7-s + 3.12·11-s − 6.56·13-s − 4.12·17-s + 5.12·19-s + 3.68·23-s − 2.56·25-s + 5.43·29-s + 2.56·31-s − 1.56·35-s + 3.56·37-s − 11.5·41-s − 9.24·43-s + 3.56·47-s + 49-s + 5.68·53-s − 4.87·55-s − 0.123·59-s + 8.24·61-s + 10.2·65-s − 15.6·67-s + 2.56·71-s + 1.12·73-s + 3.12·77-s − 4.43·79-s + 1.56·83-s + ⋯ |
| L(s) = 1 | − 0.698·5-s + 0.377·7-s + 0.941·11-s − 1.81·13-s − 0.999·17-s + 1.17·19-s + 0.768·23-s − 0.512·25-s + 1.00·29-s + 0.460·31-s − 0.263·35-s + 0.585·37-s − 1.80·41-s − 1.41·43-s + 0.519·47-s + 0.142·49-s + 0.780·53-s − 0.657·55-s − 0.0160·59-s + 1.05·61-s + 1.27·65-s − 1.91·67-s + 0.304·71-s + 0.131·73-s + 0.355·77-s − 0.499·79-s + 0.171·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 6.56T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 - 3.56T + 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 + 0.123T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 4.43T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.63292790909831278025212069472, −7.05327003267082296599275194264, −6.55390925413442249522444153366, −5.37074666119075625883512125789, −4.77635569414471297549996406512, −4.16723405678429839411115725631, −3.21062348569067196964645614477, −2.39000379768590684286381703799, −1.27012914838403455209423860209, 0,
1.27012914838403455209423860209, 2.39000379768590684286381703799, 3.21062348569067196964645614477, 4.16723405678429839411115725631, 4.77635569414471297549996406512, 5.37074666119075625883512125789, 6.55390925413442249522444153366, 7.05327003267082296599275194264, 7.63292790909831278025212069472
