L(s) = 1 | + 2.36·3-s + 2.42·5-s + 2.60·9-s − 5.39·11-s + 6.20·13-s + 5.74·15-s − 2.01·17-s + 6.62·19-s + 1.84·23-s + 0.877·25-s − 0.933·27-s + 6.47·29-s − 8.80·31-s − 12.7·33-s + 3.09·37-s + 14.6·39-s − 41-s + 10.5·43-s + 6.31·45-s + 3.78·47-s − 4.77·51-s − 1.87·53-s − 13.0·55-s + 15.6·57-s + 3.79·59-s + 6.72·61-s + 15.0·65-s + ⋯ |
L(s) = 1 | + 1.36·3-s + 1.08·5-s + 0.868·9-s − 1.62·11-s + 1.71·13-s + 1.48·15-s − 0.488·17-s + 1.52·19-s + 0.385·23-s + 0.175·25-s − 0.179·27-s + 1.20·29-s − 1.58·31-s − 2.22·33-s + 0.508·37-s + 2.35·39-s − 0.156·41-s + 1.61·43-s + 0.941·45-s + 0.552·47-s − 0.668·51-s − 0.257·53-s − 1.76·55-s + 2.07·57-s + 0.494·59-s + 0.861·61-s + 1.86·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.501343580\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.501343580\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 2.36T + 3T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 + 2.01T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 3.79T + 59T^{2} \) |
| 61 | \( 1 - 6.72T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 1.71T + 73T^{2} \) |
| 79 | \( 1 + 0.277T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 + 1.19T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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
−7.84663536865244553763916871192, −7.42175194873781820619111066045, −6.42897339421765789728057666162, −5.64931226258010244691283327395, −5.22576357754608495648956002370, −4.06486912435668663164027460280, −3.29124646302107825135274557209, −2.65467850992013031183798920096, −2.02488135188541447097495659075, −1.02014045120410202561911441466,
1.02014045120410202561911441466, 2.02488135188541447097495659075, 2.65467850992013031183798920096, 3.29124646302107825135274557209, 4.06486912435668663164027460280, 5.22576357754608495648956002370, 5.64931226258010244691283327395, 6.42897339421765789728057666162, 7.42175194873781820619111066045, 7.84663536865244553763916871192