| L(s) = 1 | + 3.66i·5-s − 7-s − 4.45i·11-s − 1.51i·13-s − 3.45·17-s + 2.29i·19-s + 8.76·23-s − 8.44·25-s − 1.62i·29-s + 9.26·31-s − 3.66i·35-s + 5.69i·37-s − 6.86·41-s + 0.880i·43-s − 10.5·47-s + ⋯ |
| L(s) = 1 | + 1.63i·5-s − 0.377·7-s − 1.34i·11-s − 0.420i·13-s − 0.837·17-s + 0.527i·19-s + 1.82·23-s − 1.68·25-s − 0.300i·29-s + 1.66·31-s − 0.619i·35-s + 0.936i·37-s − 1.07·41-s + 0.134i·43-s − 1.54·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003831294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003831294\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.66iT - 5T^{2} \) |
| 11 | \( 1 + 4.45iT - 11T^{2} \) |
| 13 | \( 1 + 1.51iT - 13T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 - 8.76T + 23T^{2} \) |
| 29 | \( 1 + 1.62iT - 29T^{2} \) |
| 31 | \( 1 - 9.26T + 31T^{2} \) |
| 37 | \( 1 - 5.69iT - 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 - 0.880iT - 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 2.99iT - 59T^{2} \) |
| 61 | \( 1 - 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 7.85T + 73T^{2} \) |
| 79 | \( 1 + 4.57T + 79T^{2} \) |
| 83 | \( 1 + 2.60iT - 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.379694891712599598844130342383, −7.56320118879178444157598264447, −6.79929712708923047601110714743, −6.38490018289435968097613095500, −5.77007359878026600564853256130, −4.75785417984127021433987718996, −3.72441610103068097077438774717, −2.87532843171269078729704643972, −2.79810351582421021456882381820, −1.17921749321237595744301752593,
0.27056365220982261768726980441, 1.39301492693064228055908485760, 2.20526967226756551406784048335, 3.35234132266219115910377936275, 4.48132984457218947732232457919, 4.76502909976467373765067201725, 5.33873016758857613662967626568, 6.61169064720030160906777200995, 6.87786874117154897002481374243, 7.911395547063765761832474269409
