| L(s) = 1 | + 2-s + 4-s − 5-s + 5.11i·7-s + 8-s − 10-s + 1.54·11-s + 2.15i·13-s + 5.11i·14-s + 16-s + 6.41i·17-s + 6.75·19-s − 20-s + 1.54·22-s + 4.83i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.93i·7-s + 0.353·8-s − 0.316·10-s + 0.466·11-s + 0.597i·13-s + 1.36i·14-s + 0.250·16-s + 1.55i·17-s + 1.55·19-s − 0.223·20-s + 0.330·22-s + 1.00i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.802225211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.802225211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-2.97 + 7.62i)T \) |
| good | 7 | \( 1 - 5.11iT - 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 - 2.15iT - 13T^{2} \) |
| 17 | \( 1 - 6.41iT - 17T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 - 4.83iT - 23T^{2} \) |
| 29 | \( 1 + 0.330iT - 29T^{2} \) |
| 31 | \( 1 + 1.13iT - 31T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 + 5.54T + 41T^{2} \) |
| 43 | \( 1 + 8.54iT - 43T^{2} \) |
| 47 | \( 1 + 4.15iT - 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 4.92iT - 59T^{2} \) |
| 61 | \( 1 + 8.79iT - 61T^{2} \) |
| 71 | \( 1 - 15.0iT - 71T^{2} \) |
| 73 | \( 1 - 7.06T + 73T^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 + 15.1iT - 83T^{2} \) |
| 89 | \( 1 - 2.94iT - 89T^{2} \) |
| 97 | \( 1 - 17.4iT - 97T^{2} \) |
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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
−8.219802365207162191971862458198, −7.67609969971710303691445691322, −6.65367853595046697658016619042, −6.08898130086048665946325491653, −5.43889631712692722037672901481, −4.86178432273388186659962792248, −3.76498240865552873299522066936, −3.27710896743458874946805910920, −2.23371803166498328351845387466, −1.51587961478161171585036942022,
0.57624313185614307491971381242, 1.29815167141310847809109664658, 2.94262295966809049815220789549, 3.32182573144079527301554708889, 4.34168937326134937581519923260, 4.67087249711379322554001399204, 5.54831463087929021679532805206, 6.68588561517887629740478527333, 6.97930413292422888144447774986, 7.73764683649681652864895458681
