| L(s) = 1 | + 2-s + 0.987·3-s + 4-s − 0.211·5-s + 0.987·6-s + 4.46·7-s + 8-s − 2.02·9-s − 0.211·10-s − 6.30·11-s + 0.987·12-s − 4.97·13-s + 4.46·14-s − 0.209·15-s + 16-s + 1.07·17-s − 2.02·18-s − 4.90·19-s − 0.211·20-s + 4.40·21-s − 6.30·22-s + 1.04·23-s + 0.987·24-s − 4.95·25-s − 4.97·26-s − 4.96·27-s + 4.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.569·3-s + 0.5·4-s − 0.0946·5-s + 0.403·6-s + 1.68·7-s + 0.353·8-s − 0.675·9-s − 0.0669·10-s − 1.89·11-s + 0.284·12-s − 1.37·13-s + 1.19·14-s − 0.0539·15-s + 0.250·16-s + 0.261·17-s − 0.477·18-s − 1.12·19-s − 0.0473·20-s + 0.961·21-s − 1.34·22-s + 0.218·23-s + 0.201·24-s − 0.991·25-s − 0.974·26-s − 0.954·27-s + 0.843·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 0.987T + 3T^{2} \) |
| 5 | \( 1 + 0.211T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 + 6.30T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 8.54T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 + 0.276T + 41T^{2} \) |
| 43 | \( 1 + 3.44T + 43T^{2} \) |
| 47 | \( 1 - 0.0554T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 1.37T + 71T^{2} \) |
| 73 | \( 1 + 2.10T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 + 1.45T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.79011747456609069877951995516, −7.33547479851520205322432243887, −6.03722716488365389829288749738, −5.40603939862838892888797881548, −4.78461829474008984490178388244, −4.30364994056749075293427491647, −3.00443568974264318467397463325, −2.46286976725664945801720038875, −1.81348846893562176114404471318, 0,
1.81348846893562176114404471318, 2.46286976725664945801720038875, 3.00443568974264318467397463325, 4.30364994056749075293427491647, 4.78461829474008984490178388244, 5.40603939862838892888797881548, 6.03722716488365389829288749738, 7.33547479851520205322432243887, 7.79011747456609069877951995516
