| L(s) = 1 | + 1.30·2-s − 3-s − 0.302·4-s + 4.30·5-s − 1.30·6-s − 3·8-s + 9-s + 5.60·10-s − 5·11-s + 0.302·12-s + 1.30·13-s − 4.30·15-s − 3.30·16-s − 1.60·17-s + 1.30·18-s − 5.60·19-s − 1.30·20-s − 6.51·22-s + 23-s + 3·24-s + 13.5·25-s + 1.69·26-s − 27-s − 8.21·29-s − 5.60·30-s − 3·31-s + 1.69·32-s + ⋯ |
| L(s) = 1 | + 0.921·2-s − 0.577·3-s − 0.151·4-s + 1.92·5-s − 0.531·6-s − 1.06·8-s + 0.333·9-s + 1.77·10-s − 1.50·11-s + 0.0874·12-s + 0.361·13-s − 1.11·15-s − 0.825·16-s − 0.389·17-s + 0.307·18-s − 1.28·19-s − 0.291·20-s − 1.38·22-s + 0.208·23-s + 0.612·24-s + 2.70·25-s + 0.332·26-s − 0.192·27-s − 1.52·29-s − 1.02·30-s − 0.538·31-s + 0.300·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 - 6.90T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 71 | \( 1 + 9.90T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 + 0.0916T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.514438048126574516222532215940, −7.09868565878067682651738535718, −6.43132888615884340089352717156, −5.64717514797860684514395956554, −5.37119402719705524147271311114, −4.71124152447525796083686680208, −3.57595384745019470383636489028, −2.50717594418137143697826850349, −1.78880239625634255939639202840, 0,
1.78880239625634255939639202840, 2.50717594418137143697826850349, 3.57595384745019470383636489028, 4.71124152447525796083686680208, 5.37119402719705524147271311114, 5.64717514797860684514395956554, 6.43132888615884340089352717156, 7.09868565878067682651738535718, 8.514438048126574516222532215940
