| L(s) = 1 | + 2-s + 3.09·3-s + 4-s + 3.65·5-s + 3.09·6-s − 3.22·7-s + 8-s + 6.54·9-s + 3.65·10-s − 1.79·11-s + 3.09·12-s − 1.26·13-s − 3.22·14-s + 11.2·15-s + 16-s + 3.88·17-s + 6.54·18-s − 3.01·19-s + 3.65·20-s − 9.97·21-s − 1.79·22-s + 6.52·23-s + 3.09·24-s + 8.35·25-s − 1.26·26-s + 10.9·27-s − 3.22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.78·3-s + 0.5·4-s + 1.63·5-s + 1.26·6-s − 1.22·7-s + 0.353·8-s + 2.18·9-s + 1.15·10-s − 0.542·11-s + 0.892·12-s − 0.351·13-s − 0.862·14-s + 2.91·15-s + 0.250·16-s + 0.943·17-s + 1.54·18-s − 0.691·19-s + 0.817·20-s − 2.17·21-s − 0.383·22-s + 1.36·23-s + 0.630·24-s + 1.67·25-s − 0.248·26-s + 2.11·27-s − 0.610·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.992035463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.992035463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 + 3.01T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 - 7.76T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 - 9.05T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 2.18T + 59T^{2} \) |
| 61 | \( 1 + 2.50T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 - 0.357T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 6.65T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 7.79T + 97T^{2} \) |
| 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
−8.491598677150910712829861023201, −7.71590321345711382775187657285, −6.86190740285279312427800053919, −6.33049643086032932196462093991, −5.44127725712453011127460773903, −4.61805201118996674659074455785, −3.44638739771146439162459793691, −2.95051278243519938225667633234, −2.36117618541226391475617954568, −1.46762742999070940006426052238,
1.46762742999070940006426052238, 2.36117618541226391475617954568, 2.95051278243519938225667633234, 3.44638739771146439162459793691, 4.61805201118996674659074455785, 5.44127725712453011127460773903, 6.33049643086032932196462093991, 6.86190740285279312427800053919, 7.71590321345711382775187657285, 8.491598677150910712829861023201
