| L(s) = 1 | + 2.99·5-s − i·7-s + 5.36i·11-s + 4.55i·13-s − 0.958i·17-s − 0.532·19-s − 2.78·23-s + 3.98·25-s + 5.15·29-s + 4.66i·31-s − 2.99i·35-s − 6.10i·37-s + 12.3i·41-s − 4.45·43-s − 1.40·47-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377i·7-s + 1.61i·11-s + 1.26i·13-s − 0.232i·17-s − 0.122·19-s − 0.581·23-s + 0.796·25-s + 0.957·29-s + 0.837i·31-s − 0.506i·35-s − 1.00i·37-s + 1.92i·41-s − 0.679·43-s − 0.204·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.161595780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.161595780\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.99T + 5T^{2} \) |
| 11 | \( 1 - 5.36iT - 11T^{2} \) |
| 13 | \( 1 - 4.55iT - 13T^{2} \) |
| 17 | \( 1 + 0.958iT - 17T^{2} \) |
| 19 | \( 1 + 0.532T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 - 4.66iT - 31T^{2} \) |
| 37 | \( 1 + 6.10iT - 37T^{2} \) |
| 41 | \( 1 - 12.3iT - 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 - 9.57T + 53T^{2} \) |
| 59 | \( 1 - 0.0356iT - 59T^{2} \) |
| 61 | \( 1 + 5.13iT - 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 + 1.90iT - 79T^{2} \) |
| 83 | \( 1 - 3.13iT - 83T^{2} \) |
| 89 | \( 1 + 5.91iT - 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.277949488131681618373461156922, −7.39097208445382348847142162143, −6.71777776188891875773217950409, −6.34073244309711289888454695887, −5.33344020735079452794577165221, −4.65832704388081331636412144802, −4.06776974500403376533744816542, −2.77582935099301762956364191826, −1.97859064588039362045976005460, −1.40261656667593405839577957726,
0.50881602016932560952981729567, 1.60909861293463993074218214975, 2.63083902808365771249140418368, 3.17360710064668722229919400169, 4.22761468770874143019794524195, 5.47356442158999619914278585794, 5.64624272178870965444499070434, 6.22118536951370206179591198215, 7.06716560493345098624619879396, 8.221161859789931289300821676997
