L(s) = 1 | − 2-s + 4-s + 3.52·5-s − 0.552·7-s − 8-s − 3.52·10-s − 4.67·11-s + 4.98·13-s + 0.552·14-s + 16-s + 5.42·17-s + 5.81·19-s + 3.52·20-s + 4.67·22-s − 1.00·23-s + 7.41·25-s − 4.98·26-s − 0.552·28-s − 1.30·29-s − 2.55·31-s − 32-s − 5.42·34-s − 1.94·35-s + 6.51·37-s − 5.81·38-s − 3.52·40-s + 1.43·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.57·5-s − 0.208·7-s − 0.353·8-s − 1.11·10-s − 1.41·11-s + 1.38·13-s + 0.147·14-s + 0.250·16-s + 1.31·17-s + 1.33·19-s + 0.787·20-s + 0.997·22-s − 0.210·23-s + 1.48·25-s − 0.977·26-s − 0.104·28-s − 0.241·29-s − 0.458·31-s − 0.176·32-s − 0.930·34-s − 0.328·35-s + 1.07·37-s − 0.943·38-s − 0.557·40-s + 0.224·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.981444404\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981444404\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 + 0.552T + 7T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 - 5.81T + 19T^{2} \) |
| 23 | \( 1 + 1.00T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 + 2.12T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 1.22T + 73T^{2} \) |
| 79 | \( 1 - 7.86T + 79T^{2} \) |
| 83 | \( 1 - 1.03T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.431012115891253695023038122019, −7.86045991110388855330838761473, −7.05082362468888923147384513916, −6.07744940278611826599769921705, −5.68454614558465918116679158288, −5.06260500474489460378899734070, −3.49146698341575216169158487000, −2.79917806950955075894138892031, −1.81325517058211431969361441153, −0.945424236987175721142637123696,
0.945424236987175721142637123696, 1.81325517058211431969361441153, 2.79917806950955075894138892031, 3.49146698341575216169158487000, 5.06260500474489460378899734070, 5.68454614558465918116679158288, 6.07744940278611826599769921705, 7.05082362468888923147384513916, 7.86045991110388855330838761473, 8.431012115891253695023038122019