| L(s) = 1 | + 3.55·5-s − 2.20·7-s + 4.03·11-s + 0.294·13-s − 7.93·17-s + 1.66·19-s − 4.24·23-s + 7.60·25-s + 5.37·29-s − 5.91·31-s − 7.83·35-s + 6.66·37-s − 6.87·41-s + 11.6·43-s + 11.6·47-s − 2.12·49-s + 1.71·53-s + 14.3·55-s + 7.53·59-s + 2.37·61-s + 1.04·65-s + 8.92·67-s + 14.2·71-s + 11.0·73-s − 8.90·77-s + 8.32·79-s − 0.974·83-s + ⋯ |
| L(s) = 1 | + 1.58·5-s − 0.834·7-s + 1.21·11-s + 0.0817·13-s − 1.92·17-s + 0.382·19-s − 0.884·23-s + 1.52·25-s + 0.997·29-s − 1.06·31-s − 1.32·35-s + 1.09·37-s − 1.07·41-s + 1.78·43-s + 1.70·47-s − 0.303·49-s + 0.235·53-s + 1.93·55-s + 0.981·59-s + 0.303·61-s + 0.129·65-s + 1.08·67-s + 1.69·71-s + 1.28·73-s − 1.01·77-s + 0.936·79-s − 0.106·83-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
| \(L(1)\) |
\(\approx\) |
\(2.597595332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.597595332\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 - 0.294T + 13T^{2} \) |
| 17 | \( 1 + 7.93T + 17T^{2} \) |
| 19 | \( 1 - 1.66T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 - 6.66T + 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 + 0.974T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.263949098769178200037176750981, −6.96733113502604004017310369634, −6.63602156595120005779475462557, −6.04953472052185401980559734968, −5.42175815025592909399519136097, −4.38098228629875584082991168729, −3.70042275467494059142018303663, −2.50671920993132094752376278308, −2.03101942950646981506686331697, −0.852553062508419378408320874810,
0.852553062508419378408320874810, 2.03101942950646981506686331697, 2.50671920993132094752376278308, 3.70042275467494059142018303663, 4.38098228629875584082991168729, 5.42175815025592909399519136097, 6.04953472052185401980559734968, 6.63602156595120005779475462557, 6.96733113502604004017310369634, 8.263949098769178200037176750981
