L(s) = 1 | − 0.539·5-s + 1.07·7-s − 3.70·11-s + 4.34·13-s + 1.46·17-s + 3.41·19-s − 8.68·23-s − 4.70·25-s − 2.15·29-s − 1.26·31-s − 0.581·35-s + 11.7·37-s − 9.95·41-s − 9.80·43-s + 7.70·47-s − 5.83·49-s + 9.55·53-s + 2·55-s − 6.92·59-s + 1.70·61-s − 2.34·65-s − 0.879·67-s − 4·71-s + 1.41·73-s − 4·77-s − 15.8·79-s + 0.183·83-s + ⋯ |
L(s) = 1 | − 0.241·5-s + 0.407·7-s − 1.11·11-s + 1.20·13-s + 0.354·17-s + 0.784·19-s − 1.80·23-s − 0.941·25-s − 0.400·29-s − 0.226·31-s − 0.0982·35-s + 1.93·37-s − 1.55·41-s − 1.49·43-s + 1.12·47-s − 0.833·49-s + 1.31·53-s + 0.269·55-s − 0.901·59-s + 0.218·61-s − 0.290·65-s − 0.107·67-s − 0.474·71-s + 0.166·73-s − 0.455·77-s − 1.78·79-s + 0.0201·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 0.539T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 + 2.15T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 9.95T + 41T^{2} \) |
| 43 | \( 1 + 9.80T + 43T^{2} \) |
| 47 | \( 1 - 7.70T + 47T^{2} \) |
| 53 | \( 1 - 9.55T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 0.879T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 0.183T + 83T^{2} \) |
| 89 | \( 1 + 6.15T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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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
−7.912193074659716619030963446180, −7.19069949958222107642056482055, −6.07535659185797601654896217454, −5.71545693289591363208641375503, −4.84708064954987266911537640589, −3.97544583953630139217965415944, −3.33035234859659137801959153193, −2.28197009010112592803274883800, −1.36162988100844992625867441643, 0,
1.36162988100844992625867441643, 2.28197009010112592803274883800, 3.33035234859659137801959153193, 3.97544583953630139217965415944, 4.84708064954987266911537640589, 5.71545693289591363208641375503, 6.07535659185797601654896217454, 7.19069949958222107642056482055, 7.912193074659716619030963446180