L(s) = 1 | + 0.860·5-s − 0.617·7-s + 4.88·11-s − 4.79·13-s + 2.56·17-s + 0.574·19-s − 1.04·23-s − 4.25·25-s − 3.36·29-s − 2.95·31-s − 0.531·35-s − 6.11·37-s − 9.79·41-s − 5.38·43-s − 6.46·47-s − 6.61·49-s − 2.44·53-s + 4.20·55-s + 1.74·59-s + 10.6·61-s − 4.12·65-s + 7.38·67-s + 7.40·71-s + 11.5·73-s − 3.01·77-s − 0.281·79-s − 8.62·83-s + ⋯ |
L(s) = 1 | + 0.384·5-s − 0.233·7-s + 1.47·11-s − 1.33·13-s + 0.622·17-s + 0.131·19-s − 0.217·23-s − 0.851·25-s − 0.625·29-s − 0.530·31-s − 0.0897·35-s − 1.00·37-s − 1.53·41-s − 0.820·43-s − 0.942·47-s − 0.945·49-s − 0.336·53-s + 0.566·55-s + 0.227·59-s + 1.36·61-s − 0.511·65-s + 0.901·67-s + 0.879·71-s + 1.35·73-s − 0.343·77-s − 0.0317·79-s − 0.946·83-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 | \( 1 - 0.860T + 5T^{2} \) |
| 7 | \( 1 + 0.617T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 0.574T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 + 3.36T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 6.11T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 - 1.74T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 0.281T + 79T^{2} \) |
| 83 | \( 1 + 8.62T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 9.14T + 97T^{2} \) |
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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
−7.71744988779481319457772926932, −6.81332964384565491683066646145, −6.54040136728834316659580487677, −5.45481348182492027373913057446, −5.00934825561018341152557470704, −3.87949299796360626501788189372, −3.38235769164523979968019375881, −2.18135146191781478054938135504, −1.46753509605523472442472427598, 0,
1.46753509605523472442472427598, 2.18135146191781478054938135504, 3.38235769164523979968019375881, 3.87949299796360626501788189372, 5.00934825561018341152557470704, 5.45481348182492027373913057446, 6.54040136728834316659580487677, 6.81332964384565491683066646145, 7.71744988779481319457772926932