| L(s) = 1 | + 3.09·3-s − 2.24·5-s + 0.244·7-s + 6.56·9-s + 2.99·11-s − 4.71·13-s − 6.93·15-s − 5.01·17-s + 2.62·19-s + 0.755·21-s − 6.39·23-s + 0.0306·25-s + 11.0·27-s − 9.36·29-s − 8.29·31-s + 9.25·33-s − 0.548·35-s − 1.92·37-s − 14.5·39-s + 8.01·41-s − 10.4·43-s − 14.7·45-s + 1.23·47-s − 6.94·49-s − 15.5·51-s − 6.99·53-s − 6.71·55-s + ⋯ |
| L(s) = 1 | + 1.78·3-s − 1.00·5-s + 0.0923·7-s + 2.18·9-s + 0.902·11-s − 1.30·13-s − 1.79·15-s − 1.21·17-s + 0.602·19-s + 0.164·21-s − 1.33·23-s + 0.00612·25-s + 2.12·27-s − 1.73·29-s − 1.49·31-s + 1.61·33-s − 0.0926·35-s − 0.316·37-s − 2.33·39-s + 1.25·41-s − 1.58·43-s − 2.19·45-s + 0.180·47-s − 0.991·49-s − 2.17·51-s − 0.961·53-s − 0.904·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 - 0.244T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 2.11T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 9.67T + 97T^{2} \) |
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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
−7.86761176734962130406505273315, −7.21591152606362967844067414947, −6.76923846245635315231774802824, −5.43420760029053366424762540790, −4.40100824561524926072897768759, −3.88522162867225834550401385102, −3.38615160964883490665304985967, −2.28729264822151119596420031883, −1.76457805785714624499220691706, 0,
1.76457805785714624499220691706, 2.28729264822151119596420031883, 3.38615160964883490665304985967, 3.88522162867225834550401385102, 4.40100824561524926072897768759, 5.43420760029053366424762540790, 6.76923846245635315231774802824, 7.21591152606362967844067414947, 7.86761176734962130406505273315
