| L(s) = 1 | + 9·5-s + 7·7-s + 6·11-s − 28·13-s − 99·17-s − 130·19-s − 102·23-s − 44·25-s + 102·29-s + 140·31-s + 63·35-s + 101·37-s − 165·41-s − 91·43-s − 111·47-s + 49·49-s − 66·53-s + 54·55-s − 675·59-s − 394·61-s − 252·65-s + 212·67-s + 48·71-s + 674·73-s + 42·77-s + 953·79-s − 825·83-s + ⋯ |
| L(s) = 1 | + 0.804·5-s + 0.377·7-s + 0.164·11-s − 0.597·13-s − 1.41·17-s − 1.56·19-s − 0.924·23-s − 0.351·25-s + 0.653·29-s + 0.811·31-s + 0.304·35-s + 0.448·37-s − 0.628·41-s − 0.322·43-s − 0.344·47-s + 1/7·49-s − 0.171·53-s + 0.132·55-s − 1.48·59-s − 0.826·61-s − 0.480·65-s + 0.386·67-s + 0.0802·71-s + 1.08·73-s + 0.0621·77-s + 1.35·79-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 99 T + p^{3} T^{2} \) |
| 19 | \( 1 + 130 T + p^{3} T^{2} \) |
| 23 | \( 1 + 102 T + p^{3} T^{2} \) |
| 29 | \( 1 - 102 T + p^{3} T^{2} \) |
| 31 | \( 1 - 140 T + p^{3} T^{2} \) |
| 37 | \( 1 - 101 T + p^{3} T^{2} \) |
| 41 | \( 1 + 165 T + p^{3} T^{2} \) |
| 43 | \( 1 + 91 T + p^{3} T^{2} \) |
| 47 | \( 1 + 111 T + p^{3} T^{2} \) |
| 53 | \( 1 + 66 T + p^{3} T^{2} \) |
| 59 | \( 1 + 675 T + p^{3} T^{2} \) |
| 61 | \( 1 + 394 T + p^{3} T^{2} \) |
| 67 | \( 1 - 212 T + p^{3} T^{2} \) |
| 71 | \( 1 - 48 T + p^{3} T^{2} \) |
| 73 | \( 1 - 674 T + p^{3} T^{2} \) |
| 79 | \( 1 - 953 T + p^{3} T^{2} \) |
| 83 | \( 1 + 825 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1398 T + p^{3} T^{2} \) |
| 97 | \( 1 + 322 T + p^{3} T^{2} \) |
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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
−9.575197550105993440936555660942, −8.682077504610946740244280887540, −7.929271239918961335655066992660, −6.65047028198015196929223673436, −6.15036322155678339463071960070, −4.90803757471799856931851179596, −4.15454593702133487369861583874, −2.53717806063403436477013586251, −1.75958403330570453532655043657, 0,
1.75958403330570453532655043657, 2.53717806063403436477013586251, 4.15454593702133487369861583874, 4.90803757471799856931851179596, 6.15036322155678339463071960070, 6.65047028198015196929223673436, 7.929271239918961335655066992660, 8.682077504610946740244280887540, 9.575197550105993440936555660942
