| L(s) = 1 | + 1.59·5-s + 1.07·7-s + 0.137·11-s − 6.04·13-s + 5.68·17-s + 19-s − 3.37·23-s − 2.45·25-s + 0.161·29-s − 8.61·31-s + 1.71·35-s + 5.51·37-s − 6.86·41-s − 4.03·43-s + 7.19·47-s − 5.84·49-s − 2.17·53-s + 0.219·55-s − 11.9·59-s + 9.67·61-s − 9.65·65-s − 10.4·67-s − 8.48·71-s + 16.3·73-s + 0.147·77-s − 8.80·79-s − 0.617·83-s + ⋯ |
| L(s) = 1 | + 0.714·5-s + 0.406·7-s + 0.0414·11-s − 1.67·13-s + 1.37·17-s + 0.229·19-s − 0.703·23-s − 0.490·25-s + 0.0300·29-s − 1.54·31-s + 0.290·35-s + 0.906·37-s − 1.07·41-s − 0.614·43-s + 1.04·47-s − 0.834·49-s − 0.298·53-s + 0.0295·55-s − 1.55·59-s + 1.23·61-s − 1.19·65-s − 1.27·67-s − 1.00·71-s + 1.90·73-s + 0.0168·77-s − 0.990·79-s − 0.0677·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 - 0.137T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 - 0.161T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 7.19T + 47T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 9.67T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 + 0.617T + 83T^{2} \) |
| 89 | \( 1 + 3.11T + 89T^{2} \) |
| 97 | \( 1 - 1.20T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69183272694361192244244984324, −6.82647668328674811452629676117, −5.94632587757781022225443026411, −5.38207959071256925891396427355, −4.83945619492147397871441982185, −3.89071278423641042873032300112, −2.99991805698079417663353362731, −2.14698276903345732806118550751, −1.43643721780858356991913167617, 0,
1.43643721780858356991913167617, 2.14698276903345732806118550751, 2.99991805698079417663353362731, 3.89071278423641042873032300112, 4.83945619492147397871441982185, 5.38207959071256925891396427355, 5.94632587757781022225443026411, 6.82647668328674811452629676117, 7.69183272694361192244244984324
