| L(s) = 1 | + 2-s − 0.399·3-s + 4-s − 0.854·5-s − 0.399·6-s − 0.419·7-s + 8-s − 2.84·9-s − 0.854·10-s + 5.34·11-s − 0.399·12-s − 0.793·13-s − 0.419·14-s + 0.341·15-s + 16-s + 4.29·17-s − 2.84·18-s − 5.96·19-s − 0.854·20-s + 0.167·21-s + 5.34·22-s − 5.69·23-s − 0.399·24-s − 4.27·25-s − 0.793·26-s + 2.33·27-s − 0.419·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.230·3-s + 0.5·4-s − 0.381·5-s − 0.163·6-s − 0.158·7-s + 0.353·8-s − 0.946·9-s − 0.270·10-s + 1.61·11-s − 0.115·12-s − 0.220·13-s − 0.112·14-s + 0.0881·15-s + 0.250·16-s + 1.04·17-s − 0.669·18-s − 1.36·19-s − 0.190·20-s + 0.0366·21-s + 1.14·22-s − 1.18·23-s − 0.0816·24-s − 0.854·25-s − 0.155·26-s + 0.449·27-s − 0.0793·28-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 | \( 1 + 0.399T + 3T^{2} \) |
| 5 | \( 1 + 0.854T + 5T^{2} \) |
| 7 | \( 1 + 0.419T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 0.793T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 + 0.739T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 + 7.06T + 41T^{2} \) |
| 43 | \( 1 - 2.16T + 43T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 2.67T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 + 3.53T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 + 2.16T + 83T^{2} \) |
| 89 | \( 1 + 6.14T + 89T^{2} \) |
| 97 | \( 1 - 7.17T + 97T^{2} \) |
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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
−7.79077412444186227339887573025, −6.74901135022166628689334208433, −6.16874193977437474961565468193, −5.76681332398546095321146771825, −4.72675647310170922733047040790, −3.97884647889616656511671734748, −3.48982814895837417377292610773, −2.44767194310570674036345015419, −1.45263575361054992768218596043, 0,
1.45263575361054992768218596043, 2.44767194310570674036345015419, 3.48982814895837417377292610773, 3.97884647889616656511671734748, 4.72675647310170922733047040790, 5.76681332398546095321146771825, 6.16874193977437474961565468193, 6.74901135022166628689334208433, 7.79077412444186227339887573025
