| L(s) = 1 | + 2-s + 4-s + 3.16·5-s + 3.68·7-s + 8-s + 3.16·10-s − 2.28·11-s + 3.10·13-s + 3.68·14-s + 16-s + 1.72·17-s − 3.39·19-s + 3.16·20-s − 2.28·22-s − 3.35·23-s + 5.03·25-s + 3.10·26-s + 3.68·28-s − 0.585·29-s − 4.63·31-s + 32-s + 1.72·34-s + 11.6·35-s − 7.30·37-s − 3.39·38-s + 3.16·40-s − 7.08·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.41·5-s + 1.39·7-s + 0.353·8-s + 1.00·10-s − 0.688·11-s + 0.859·13-s + 0.985·14-s + 0.250·16-s + 0.418·17-s − 0.778·19-s + 0.708·20-s − 0.487·22-s − 0.699·23-s + 1.00·25-s + 0.608·26-s + 0.696·28-s − 0.108·29-s − 0.833·31-s + 0.176·32-s + 0.295·34-s + 1.97·35-s − 1.20·37-s − 0.550·38-s + 0.500·40-s − 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.811449073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.811449073\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 + 0.585T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + 7.30T + 37T^{2} \) |
| 41 | \( 1 + 7.08T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 3.85T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 8.60T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 3.09T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 9.19T + 97T^{2} \) |
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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
−9.632349058028414735737080082344, −8.574777415534822861632506935199, −7.983328829102067237770648953315, −6.89020362706109351804814305253, −5.96206397737256696236764635277, −5.39071929019863982082645026147, −4.68405931517376159579123545426, −3.50278265581209543724892318988, −2.15941065080401216131520827169, −1.60025427704436387408655643684,
1.60025427704436387408655643684, 2.15941065080401216131520827169, 3.50278265581209543724892318988, 4.68405931517376159579123545426, 5.39071929019863982082645026147, 5.96206397737256696236764635277, 6.89020362706109351804814305253, 7.983328829102067237770648953315, 8.574777415534822861632506935199, 9.632349058028414735737080082344
