| L(s) = 1 | − 0.610·5-s − 0.184·7-s + 5.07·11-s + 2.65·13-s + 0.128·17-s − 7.41·19-s − 3.99·23-s − 4.62·25-s + 1.33·29-s − 8.19·31-s + 0.112·35-s + 3.63·37-s + 2.15·41-s − 2.76·43-s − 4.94·47-s − 6.96·49-s + 0.756·53-s − 3.09·55-s + 6.98·59-s − 0.248·61-s − 1.62·65-s − 15.5·67-s − 4.03·71-s − 1.57·73-s − 0.935·77-s + 8.12·79-s + 15.0·83-s + ⋯ |
| L(s) = 1 | − 0.272·5-s − 0.0696·7-s + 1.53·11-s + 0.736·13-s + 0.0311·17-s − 1.70·19-s − 0.832·23-s − 0.925·25-s + 0.247·29-s − 1.47·31-s + 0.0190·35-s + 0.598·37-s + 0.336·41-s − 0.420·43-s − 0.720·47-s − 0.995·49-s + 0.103·53-s − 0.417·55-s + 0.909·59-s − 0.0318·61-s − 0.201·65-s − 1.89·67-s − 0.478·71-s − 0.184·73-s − 0.106·77-s + 0.914·79-s + 1.65·83-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 | \( 1 + 0.610T + 5T^{2} \) |
| 7 | \( 1 + 0.184T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 - 0.128T + 17T^{2} \) |
| 19 | \( 1 + 7.41T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 - 2.15T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 0.756T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 61 | \( 1 + 0.248T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 3.67T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.83106029361793582666980566862, −6.88821471520168120216598959983, −6.29769879723395629827338020089, −5.82612918614404955561004127350, −4.62365586695248838983339801677, −3.96013905398544670872273507717, −3.50440826776738804488333113848, −2.16919090028782273055818875306, −1.40891156435649090877202780522, 0,
1.40891156435649090877202780522, 2.16919090028782273055818875306, 3.50440826776738804488333113848, 3.96013905398544670872273507717, 4.62365586695248838983339801677, 5.82612918614404955561004127350, 6.29769879723395629827338020089, 6.88821471520168120216598959983, 7.83106029361793582666980566862
