| L(s) = 1 | + 3·2-s + 4-s − 6·5-s − 28·7-s − 21·8-s − 18·10-s + 24·11-s − 58·13-s − 84·14-s − 71·16-s − 17·17-s + 116·19-s − 6·20-s + 72·22-s + 60·23-s − 89·25-s − 174·26-s − 28·28-s − 30·29-s − 172·31-s − 45·32-s − 51·34-s + 168·35-s − 58·37-s + 348·38-s + 126·40-s + 342·41-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 0.536·5-s − 1.51·7-s − 0.928·8-s − 0.569·10-s + 0.657·11-s − 1.23·13-s − 1.60·14-s − 1.10·16-s − 0.242·17-s + 1.40·19-s − 0.0670·20-s + 0.697·22-s + 0.543·23-s − 0.711·25-s − 1.31·26-s − 0.188·28-s − 0.192·29-s − 0.996·31-s − 0.248·32-s − 0.257·34-s + 0.811·35-s − 0.257·37-s + 1.48·38-s + 0.498·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 60 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 172 T + p^{3} T^{2} \) |
| 37 | \( 1 + 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 342 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 288 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 252 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 - 362 T + p^{3} T^{2} \) |
| 79 | \( 1 + 484 T + p^{3} T^{2} \) |
| 83 | \( 1 + 756 T + p^{3} T^{2} \) |
| 89 | \( 1 - 774 T + p^{3} T^{2} \) |
| 97 | \( 1 + 382 T + p^{3} T^{2} \) |
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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
−12.29832645046066653290861761855, −11.44486693547653239374504329741, −9.747235020419909570129643868121, −9.201717554277108621744918319431, −7.44341663706080716828933604201, −6.41269876808547644856589628353, −5.21651946848089499214866262266, −3.89996217873455686527651949802, −2.98219859425725005523103074571, 0,
2.98219859425725005523103074571, 3.89996217873455686527651949802, 5.21651946848089499214866262266, 6.41269876808547644856589628353, 7.44341663706080716828933604201, 9.201717554277108621744918319431, 9.747235020419909570129643868121, 11.44486693547653239374504329741, 12.29832645046066653290861761855
