| L(s) = 1 | − 2.82·7-s + 2.82·11-s − 13-s − 2.82·19-s + 5.65·23-s − 5·25-s − 4·29-s + 8.48·31-s + 6·37-s − 12·41-s + 2.82·47-s + 1.00·49-s − 4·53-s − 14.1·59-s + 6·61-s − 8.48·67-s − 8.48·71-s − 10·73-s − 8.00·77-s + 11.3·79-s + 14.1·83-s + 4·89-s + 2.82·91-s − 2·97-s − 12·101-s − 5.65·107-s + 2·109-s + ⋯ |
| L(s) = 1 | − 1.06·7-s + 0.852·11-s − 0.277·13-s − 0.648·19-s + 1.17·23-s − 25-s − 0.742·29-s + 1.52·31-s + 0.986·37-s − 1.87·41-s + 0.412·47-s + 0.142·49-s − 0.549·53-s − 1.84·59-s + 0.768·61-s − 1.03·67-s − 1.00·71-s − 1.17·73-s − 0.911·77-s + 1.27·79-s + 1.55·83-s + 0.423·89-s + 0.296·91-s − 0.203·97-s − 1.19·101-s − 0.546·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−8.148195651964430343567655986611, −7.34762876034874489524250913262, −6.47605889546831238003927906671, −6.21535049052104737626544214625, −5.09232431322293832392240091046, −4.23475542682582328790116216297, −3.42601413599798036330252233699, −2.62422901207505971174675375510, −1.40039351987246288946633728416, 0,
1.40039351987246288946633728416, 2.62422901207505971174675375510, 3.42601413599798036330252233699, 4.23475542682582328790116216297, 5.09232431322293832392240091046, 6.21535049052104737626544214625, 6.47605889546831238003927906671, 7.34762876034874489524250913262, 8.148195651964430343567655986611
