L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + 9-s − 0.999·10-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.499 + 0.866i)20-s + 0.999·26-s + (0.5 − 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + 9-s − 0.999·10-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.499 + 0.866i)20-s + 0.999·26-s + (0.5 − 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.069973285\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.069973285\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.169994380759041792596594353303, −8.370394961894453363948651735784, −7.48895299926007825561392559393, −6.88521173603733345270066226813, −5.61624032167894390090320536363, −4.63577592817786342317343909150, −4.24415461387670700211252450772, −2.94770513097684583366159477809, −1.92715743686348076457171395992, −1.03728765299495148164202882245,
1.27285184520674591537394984693, 2.47759962026846324014870392190, 3.71295729323798562955640664206, 4.74387043071975521138624765837, 5.58449089320518232905719832117, 6.36118402844847364745752330287, 6.95612315131344762548278446431, 7.69689779615188450859645286479, 8.280870944479071571542216532270, 9.372759250997129757723669293646