| L(s) = 1 | − 2.76·5-s + (0.480 + 2.60i)7-s − 5.75·11-s − 2·13-s + 6.71i·17-s + 5.20i·19-s − 4.43i·23-s + 2.66·25-s + 1.54i·29-s + 8.05·31-s + (−1.33 − 7.20i)35-s + 4.42i·37-s − 0.209i·41-s − 10.5·43-s − 4.58·47-s + ⋯ |
| L(s) = 1 | − 1.23·5-s + (0.181 + 0.983i)7-s − 1.73·11-s − 0.554·13-s + 1.62i·17-s + 1.19i·19-s − 0.924i·23-s + 0.533·25-s + 0.286i·29-s + 1.44·31-s + (−0.225 − 1.21i)35-s + 0.727i·37-s − 0.0326i·41-s − 1.60·43-s − 0.669·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0789 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0789 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1309244862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1309244862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.480 - 2.60i)T \) |
| good | 5 | \( 1 + 2.76T + 5T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 6.71iT - 17T^{2} \) |
| 19 | \( 1 - 5.20iT - 19T^{2} \) |
| 23 | \( 1 + 4.43iT - 23T^{2} \) |
| 29 | \( 1 - 1.54iT - 29T^{2} \) |
| 31 | \( 1 - 8.05T + 31T^{2} \) |
| 37 | \( 1 - 4.42iT - 37T^{2} \) |
| 41 | \( 1 + 0.209iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 4.58T + 47T^{2} \) |
| 53 | \( 1 + 8.05iT - 53T^{2} \) |
| 59 | \( 1 - 5.53iT - 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.04T + 67T^{2} \) |
| 71 | \( 1 - 1.10iT - 71T^{2} \) |
| 73 | \( 1 + 9.59iT - 73T^{2} \) |
| 79 | \( 1 + 14.7iT - 79T^{2} \) |
| 83 | \( 1 + 8.86iT - 83T^{2} \) |
| 89 | \( 1 - 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 4.58iT - 97T^{2} \) |
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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
−8.193436298317961961690382225224, −7.918853039837993494993824039282, −6.80002153588827362458634073574, −5.99906061309134282620769351063, −5.16975836231702553353025925702, −4.54474265766582908384565545680, −3.55673536363540463607721358265, −2.76853803889421822755722602690, −1.81393134748001557270330283478, −0.05294677991150152757921431471,
0.75161811497010203026987075404, 2.49479402271405981982089005889, 3.14696368614023366014264416555, 4.15379239440164883818555950802, 4.89483835751750052586449250947, 5.31837337394671025360524713341, 6.89505442424972009885553110659, 7.15532198465665684025236483484, 7.977949338035910811098319792197, 8.209043289921856757993058583868
