| L(s) = 1 | − 4·7-s − 4·11-s + 4·13-s + 2·17-s − 4·19-s + 8·23-s − 5·25-s + 8·29-s − 4·31-s − 4·37-s − 6·41-s + 4·43-s + 8·47-s + 9·49-s + 8·53-s + 12·59-s + 12·61-s + 12·67-s − 8·71-s − 6·73-s + 16·77-s − 4·79-s + 4·83-s + 6·89-s − 16·91-s − 2·97-s + 8·101-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 1.10·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 25-s + 1.48·29-s − 0.718·31-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.09·53-s + 1.56·59-s + 1.53·61-s + 1.46·67-s − 0.949·71-s − 0.702·73-s + 1.82·77-s − 0.450·79-s + 0.439·83-s + 0.635·89-s − 1.67·91-s − 0.203·97-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.254383622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254383622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.854095170137570041418708753560, −8.452848989138624536370594141032, −7.32050070381589312579630771700, −6.69832497050151318432832136976, −5.89248824651408093120164325311, −5.20378322714597588255544570785, −3.95328051751477081818950800416, −3.23090634384327317924530243659, −2.36369109468527969660219240452, −0.70235588684448499632011125436,
0.70235588684448499632011125436, 2.36369109468527969660219240452, 3.23090634384327317924530243659, 3.95328051751477081818950800416, 5.20378322714597588255544570785, 5.89248824651408093120164325311, 6.69832497050151318432832136976, 7.32050070381589312579630771700, 8.452848989138624536370594141032, 8.854095170137570041418708753560
