| L(s) = 1 | + 2-s + 4-s + 5-s − 2.60·7-s + 8-s + 10-s + 13-s − 2.60·14-s + 16-s + 4.60·17-s + 6.60·19-s + 20-s + 4.60·23-s + 25-s + 26-s − 2.60·28-s − 4.60·29-s + 2·31-s + 32-s + 4.60·34-s − 2.60·35-s + 11.2·37-s + 6.60·38-s + 40-s + 3.21·41-s + 5.21·43-s + 4.60·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.984·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s − 0.696·14-s + 0.250·16-s + 1.11·17-s + 1.51·19-s + 0.223·20-s + 0.960·23-s + 0.200·25-s + 0.196·26-s − 0.492·28-s − 0.855·29-s + 0.359·31-s + 0.176·32-s + 0.789·34-s − 0.440·35-s + 1.84·37-s + 1.07·38-s + 0.158·40-s + 0.501·41-s + 0.794·43-s + 0.679·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.686641799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.686641799\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 + 1.21T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 - 6.60T + 97T^{2} \) |
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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
−9.635986929923984894284432054030, −9.326358798062685494621007363038, −7.897317131235509701685615963060, −7.20750154291776503825351295184, −6.18339040129388593714578191135, −5.66479110705734512417140733146, −4.65590499977135051485604735436, −3.40746312614170244869524286466, −2.84591525267786939907063306852, −1.22386453611003018721553684630,
1.22386453611003018721553684630, 2.84591525267786939907063306852, 3.40746312614170244869524286466, 4.65590499977135051485604735436, 5.66479110705734512417140733146, 6.18339040129388593714578191135, 7.20750154291776503825351295184, 7.897317131235509701685615963060, 9.326358798062685494621007363038, 9.635986929923984894284432054030
