| L(s) = 1 | − 1.26·5-s − 4.29·7-s − 3.92·11-s + 4.42·13-s + 3.68·17-s − 5.02·19-s + 4.39·23-s − 3.41·25-s + 2.76·29-s + 1.68·31-s + 5.41·35-s + 10.5·37-s + 4.81·41-s + 9.63·43-s − 3.00·47-s + 11.4·49-s + 53-s + 4.95·55-s − 0.101·59-s − 14.8·61-s − 5.57·65-s + 1.35·67-s + 14.7·71-s − 9.57·73-s + 16.8·77-s − 14.7·79-s + 0.583·83-s + ⋯ |
| L(s) = 1 | − 0.563·5-s − 1.62·7-s − 1.18·11-s + 1.22·13-s + 0.893·17-s − 1.15·19-s + 0.917·23-s − 0.682·25-s + 0.513·29-s + 0.302·31-s + 0.915·35-s + 1.74·37-s + 0.752·41-s + 1.47·43-s − 0.438·47-s + 1.63·49-s + 0.137·53-s + 0.667·55-s − 0.0132·59-s − 1.90·61-s − 0.691·65-s + 0.165·67-s + 1.75·71-s − 1.12·73-s + 1.92·77-s − 1.66·79-s + 0.0640·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 1.26T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 13 | \( 1 - 4.42T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 1.68T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 - 9.63T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 59 | \( 1 + 0.101T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 1.35T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 9.57T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 0.583T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−7.60686569486752496360971104628, −6.79340337176560609419927992969, −6.04101974556905678574004331910, −5.73254856562844474743495991927, −4.51033242215875101499629563762, −3.88376949533498849723988094810, −3.06680886868476216197926825873, −2.57579863416062185286678446073, −1.02985404370263428677641485936, 0,
1.02985404370263428677641485936, 2.57579863416062185286678446073, 3.06680886868476216197926825873, 3.88376949533498849723988094810, 4.51033242215875101499629563762, 5.73254856562844474743495991927, 6.04101974556905678574004331910, 6.79340337176560609419927992969, 7.60686569486752496360971104628
