| L(s) = 1 | + 0.420·3-s + 2.86·7-s − 2.82·9-s − 3.96·11-s − 3.36·13-s + 4.80·17-s − 7.66·19-s + 1.20·21-s + 8.44·23-s − 2.44·27-s + 4.18·29-s + 6.15·31-s − 1.66·33-s + 1.44·37-s − 1.41·39-s + 1.77·41-s − 7.76·43-s − 11.0·47-s + 1.22·49-s + 2.01·51-s − 10.4·53-s − 3.22·57-s + 7.19·59-s − 14.7·61-s − 8.09·63-s + 9.39·67-s + 3.55·69-s + ⋯ |
| L(s) = 1 | + 0.242·3-s + 1.08·7-s − 0.941·9-s − 1.19·11-s − 0.932·13-s + 1.16·17-s − 1.75·19-s + 0.263·21-s + 1.76·23-s − 0.471·27-s + 0.777·29-s + 1.10·31-s − 0.290·33-s + 0.236·37-s − 0.226·39-s + 0.277·41-s − 1.18·43-s − 1.61·47-s + 0.174·49-s + 0.282·51-s − 1.43·53-s − 0.426·57-s + 0.936·59-s − 1.88·61-s − 1.02·63-s + 1.14·67-s + 0.427·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.420T + 3T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 7.66T + 19T^{2} \) |
| 23 | \( 1 - 8.44T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 - 1.77T + 41T^{2} \) |
| 43 | \( 1 + 7.76T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 9.39T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 16.7T + 73T^{2} \) |
| 79 | \( 1 + 6.42T + 79T^{2} \) |
| 83 | \( 1 + 4.07T + 83T^{2} \) |
| 89 | \( 1 + 0.854T + 89T^{2} \) |
| 97 | \( 1 + 5.04T + 97T^{2} \) |
| 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
−8.270462729541736660004375657902, −7.56282564944197137945969250133, −6.65581436484984722004902481783, −5.73446374999816631047501583419, −4.92111965620508130805772345519, −4.60716873296616728856115335870, −3.07790781335180670306451495514, −2.65543426674096998592805316737, −1.51566728520758612798854974738, 0,
1.51566728520758612798854974738, 2.65543426674096998592805316737, 3.07790781335180670306451495514, 4.60716873296616728856115335870, 4.92111965620508130805772345519, 5.73446374999816631047501583419, 6.65581436484984722004902481783, 7.56282564944197137945969250133, 8.270462729541736660004375657902
