| L(s) = 1 | + 2.85·3-s + 3.01·7-s + 5.17·9-s − 0.409·11-s − 6.00·13-s + 7.27·17-s + 0.791·19-s + 8.61·21-s + 5.27·23-s + 6.22·27-s + 29-s + 0.678·31-s − 1.17·33-s − 1.55·37-s − 17.1·39-s − 9.09·41-s − 4.32·43-s + 7.69·47-s + 2.08·49-s + 20.7·51-s + 8.35·53-s + 2.26·57-s − 9.43·59-s + 12.2·61-s + 15.6·63-s + 13.6·67-s + 15.0·69-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 1.13·7-s + 1.72·9-s − 0.123·11-s − 1.66·13-s + 1.76·17-s + 0.181·19-s + 1.88·21-s + 1.10·23-s + 1.19·27-s + 0.185·29-s + 0.121·31-s − 0.203·33-s − 0.255·37-s − 2.75·39-s − 1.41·41-s − 0.658·43-s + 1.12·47-s + 0.298·49-s + 2.91·51-s + 1.14·53-s + 0.299·57-s − 1.22·59-s + 1.56·61-s + 1.96·63-s + 1.66·67-s + 1.81·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.922931953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.922931953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.85T + 3T^{2} \) |
| 7 | \( 1 - 3.01T + 7T^{2} \) |
| 11 | \( 1 + 0.409T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 - 7.27T + 17T^{2} \) |
| 19 | \( 1 - 0.791T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 31 | \( 1 - 0.678T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 + 9.09T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 9.08T + 97T^{2} \) |
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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
−8.601168500762139982111251722792, −8.056117239319366987195393025433, −7.48620461023901016195687259092, −6.95179923050424585767177689275, −5.33918543817283004985030035392, −4.90748117322728472642576383034, −3.82626874178006870454321281406, −2.98412147992844808479611183919, −2.25659680625460993939516245743, −1.26223456700194848743907554482,
1.26223456700194848743907554482, 2.25659680625460993939516245743, 2.98412147992844808479611183919, 3.82626874178006870454321281406, 4.90748117322728472642576383034, 5.33918543817283004985030035392, 6.95179923050424585767177689275, 7.48620461023901016195687259092, 8.056117239319366987195393025433, 8.601168500762139982111251722792
