| L(s) = 1 | + 4.38·5-s − 7-s − 3.59·11-s + 0.797·13-s + 0.295·17-s − 0.704·19-s + 7.86·23-s + 14.2·25-s + 4.50·29-s + 1.50·31-s − 4.38·35-s − 0.202·37-s − 8.66·41-s + 2.79·43-s − 8.27·47-s + 49-s + 4·53-s − 15.7·55-s − 8.27·59-s + 11.9·61-s + 3.49·65-s − 13.3·67-s + 6.79·71-s + 6.20·73-s + 3.59·77-s + 14.7·79-s + 14.6·83-s + ⋯ |
| L(s) = 1 | + 1.96·5-s − 0.377·7-s − 1.08·11-s + 0.221·13-s + 0.0717·17-s − 0.161·19-s + 1.63·23-s + 2.85·25-s + 0.835·29-s + 0.269·31-s − 0.741·35-s − 0.0333·37-s − 1.35·41-s + 0.426·43-s − 1.20·47-s + 0.142·49-s + 0.549·53-s − 2.12·55-s − 1.07·59-s + 1.53·61-s + 0.433·65-s − 1.63·67-s + 0.806·71-s + 0.726·73-s + 0.409·77-s + 1.65·79-s + 1.60·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.880439366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.880439366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 4.38T + 5T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 0.797T + 13T^{2} \) |
| 17 | \( 1 - 0.295T + 17T^{2} \) |
| 19 | \( 1 + 0.704T + 19T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 + 0.202T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 + 8.27T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 8.27T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 - 6.20T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 + 8.54T + 97T^{2} \) |
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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
−8.218557751318228166737347288436, −7.16750724481053843522867271683, −6.52172033353346146329418028140, −6.01349704879142990509022691509, −5.08468204064256586465355956764, −4.92612257852334485700045618788, −3.33949491242411602768157737031, −2.69359989457237677866955755536, −1.94842759522955328127091660201, −0.910499198344626785999806309461,
0.910499198344626785999806309461, 1.94842759522955328127091660201, 2.69359989457237677866955755536, 3.33949491242411602768157737031, 4.92612257852334485700045618788, 5.08468204064256586465355956764, 6.01349704879142990509022691509, 6.52172033353346146329418028140, 7.16750724481053843522867271683, 8.218557751318228166737347288436
