| L(s) = 1 | − 5-s − 7-s + 6.24·11-s − 2.58·13-s + 1.58·17-s + 7.07·19-s + 23-s + 25-s + 7.24·29-s + 0.656·31-s + 35-s − 3·37-s + 4.41·41-s + 2·43-s − 2.58·47-s − 6·49-s + 3.58·53-s − 6.24·55-s − 2.07·59-s − 13.0·61-s + 2.58·65-s − 0.656·67-s + 12.0·71-s − 15.5·73-s − 6.24·77-s − 5.65·79-s − 1.92·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.88·11-s − 0.717·13-s + 0.384·17-s + 1.62·19-s + 0.208·23-s + 0.200·25-s + 1.34·29-s + 0.117·31-s + 0.169·35-s − 0.493·37-s + 0.689·41-s + 0.304·43-s − 0.377·47-s − 0.857·49-s + 0.492·53-s − 0.841·55-s − 0.269·59-s − 1.67·61-s + 0.320·65-s − 0.0802·67-s + 1.43·71-s − 1.82·73-s − 0.711·77-s − 0.636·79-s − 0.211·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.125907345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125907345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 - 0.656T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 - 3.58T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 0.656T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 - 5.17T + 89T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
| 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
−7.64916091622334065085584798376, −7.14792290163833693241242202052, −6.50293384632706340990334137914, −5.83319670881768815172607799934, −4.89056857067715478345244729134, −4.28309053598438319596678887160, −3.41299234227397732696887327406, −2.91416709248595091567466973578, −1.57576886792507690170880026436, −0.76911822755973249307739037935,
0.76911822755973249307739037935, 1.57576886792507690170880026436, 2.91416709248595091567466973578, 3.41299234227397732696887327406, 4.28309053598438319596678887160, 4.89056857067715478345244729134, 5.83319670881768815172607799934, 6.50293384632706340990334137914, 7.14792290163833693241242202052, 7.64916091622334065085584798376
