| L(s) = 1 | + 2.46·3-s + 2.10·5-s + 3.09·9-s − 4.68·11-s + 5.97·13-s + 5.20·15-s + 1.75·17-s + 3.07·19-s + 7.82·23-s − 0.561·25-s + 0.237·27-s − 8.57·29-s + 10.7·31-s − 11.5·33-s − 7.27·37-s + 14.7·39-s + 0.134·41-s − 2.06·43-s + 6.52·45-s + 2.55·47-s + 4.32·51-s − 6.06·53-s − 9.86·55-s + 7.59·57-s + 3.87·59-s − 3.04·61-s + 12.5·65-s + ⋯ |
| L(s) = 1 | + 1.42·3-s + 0.942·5-s + 1.03·9-s − 1.41·11-s + 1.65·13-s + 1.34·15-s + 0.424·17-s + 0.705·19-s + 1.63·23-s − 0.112·25-s + 0.0456·27-s − 1.59·29-s + 1.92·31-s − 2.01·33-s − 1.19·37-s + 2.36·39-s + 0.0210·41-s − 0.314·43-s + 0.972·45-s + 0.372·47-s + 0.605·51-s − 0.832·53-s − 1.33·55-s + 1.00·57-s + 0.504·59-s − 0.390·61-s + 1.56·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.789745709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.789745709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.46T + 3T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 11 | \( 1 + 4.68T + 11T^{2} \) |
| 13 | \( 1 - 5.97T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 - 0.134T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 - 2.55T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 - 3.87T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 + 5.33T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 8.25T + 83T^{2} \) |
| 89 | \( 1 - 4.46T + 89T^{2} \) |
| 97 | \( 1 + 0.302T + 97T^{2} \) |
| 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
−8.839324834823597570099031413658, −8.131427451445679182800798172746, −7.58989984992086520854241999783, −6.58981893794797491290591257939, −5.66590508061309560211352700261, −5.02556734394610104585189494441, −3.67621592493186343903613557616, −3.07620294878469836251930057542, −2.26100842028666206049990689574, −1.26267779713096543809348528095,
1.26267779713096543809348528095, 2.26100842028666206049990689574, 3.07620294878469836251930057542, 3.67621592493186343903613557616, 5.02556734394610104585189494441, 5.66590508061309560211352700261, 6.58981893794797491290591257939, 7.58989984992086520854241999783, 8.131427451445679182800798172746, 8.839324834823597570099031413658
