| L(s) = 1 | + 0.629·2-s − 1.60·4-s + 1.36·5-s − 0.587·7-s − 2.26·8-s + 0.860·10-s + 3.79·13-s − 0.370·14-s + 1.77·16-s + 0.193·17-s − 1.94·19-s − 2.19·20-s − 0.321·23-s − 3.13·25-s + 2.39·26-s + 0.942·28-s + 4.17·29-s + 5.63·31-s + 5.65·32-s + 0.121·34-s − 0.803·35-s + 10.2·37-s − 1.22·38-s − 3.10·40-s − 6.86·41-s + 6.81·43-s − 0.202·46-s + ⋯ |
| L(s) = 1 | + 0.445·2-s − 0.801·4-s + 0.611·5-s − 0.222·7-s − 0.802·8-s + 0.272·10-s + 1.05·13-s − 0.0989·14-s + 0.444·16-s + 0.0469·17-s − 0.446·19-s − 0.490·20-s − 0.0669·23-s − 0.626·25-s + 0.469·26-s + 0.178·28-s + 0.774·29-s + 1.01·31-s + 1.00·32-s + 0.0208·34-s − 0.135·35-s + 1.69·37-s − 0.198·38-s − 0.490·40-s − 1.07·41-s + 1.03·43-s − 0.0298·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.169835596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.169835596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.629T + 2T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 7 | \( 1 + 0.587T + 7T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 - 0.193T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 + 0.321T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 - 5.63T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 - 6.81T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 + 7.86T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 + 8.63T + 61T^{2} \) |
| 67 | \( 1 + 9.09T + 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 - 0.424T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
| 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
−7.907275000355748871313180026090, −6.68423294580760467160495173640, −6.12654972778909408252675534103, −5.76634117962301350832988744537, −4.76921881969619706437673565119, −4.33144422519519376759696083715, −3.45369420920332943903858424953, −2.81314934475870088078020577669, −1.70074557685441949057307697874, −0.67250492537011210485914002260,
0.67250492537011210485914002260, 1.70074557685441949057307697874, 2.81314934475870088078020577669, 3.45369420920332943903858424953, 4.33144422519519376759696083715, 4.76921881969619706437673565119, 5.76634117962301350832988744537, 6.12654972778909408252675534103, 6.68423294580760467160495173640, 7.907275000355748871313180026090
