| L(s) = 1 | − 2-s + 3.26·3-s + 4-s + 0.595·5-s − 3.26·6-s − 2.99·7-s − 8-s + 7.67·9-s − 0.595·10-s + 0.495·11-s + 3.26·12-s − 2.84·13-s + 2.99·14-s + 1.94·15-s + 16-s − 1.90·17-s − 7.67·18-s − 19-s + 0.595·20-s − 9.77·21-s − 0.495·22-s − 8.88·23-s − 3.26·24-s − 4.64·25-s + 2.84·26-s + 15.2·27-s − 2.99·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.88·3-s + 0.5·4-s + 0.266·5-s − 1.33·6-s − 1.13·7-s − 0.353·8-s + 2.55·9-s − 0.188·10-s + 0.149·11-s + 0.943·12-s − 0.789·13-s + 0.799·14-s + 0.502·15-s + 0.250·16-s − 0.461·17-s − 1.80·18-s − 0.229·19-s + 0.133·20-s − 2.13·21-s − 0.105·22-s − 1.85·23-s − 0.667·24-s − 0.929·25-s + 0.558·26-s + 2.94·27-s − 0.565·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 3.26T + 3T^{2} \) |
| 5 | \( 1 - 0.595T + 5T^{2} \) |
| 7 | \( 1 + 2.99T + 7T^{2} \) |
| 11 | \( 1 - 0.495T + 11T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 23 | \( 1 + 8.88T + 23T^{2} \) |
| 29 | \( 1 + 0.0817T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 - 0.933T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 + 3.28T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 3.99T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 6.88T + 97T^{2} \) |
| 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
−7.66296283021669515830550627832, −7.07260659425530615471404272732, −6.41747212101862587240939116196, −5.57246092077319608905017352056, −4.10870645929102387504647022089, −3.86981426497806290566136469578, −2.76031635301284324073435244115, −2.37961309095389551026327935975, −1.57419353894283704324174263052, 0,
1.57419353894283704324174263052, 2.37961309095389551026327935975, 2.76031635301284324073435244115, 3.86981426497806290566136469578, 4.10870645929102387504647022089, 5.57246092077319608905017352056, 6.41747212101862587240939116196, 7.07260659425530615471404272732, 7.66296283021669515830550627832
