| L(s) = 1 | + 0.618·2-s + 3-s − 1.61·4-s + 2.23·5-s + 0.618·6-s − 2.23·8-s + 9-s + 1.38·10-s + 2.23·11-s − 1.61·12-s + 6.23·13-s + 2.23·15-s + 1.85·16-s − 1.85·17-s + 0.618·18-s − 3.09·19-s − 3.61·20-s + 1.38·22-s − 4.61·23-s − 2.23·24-s + 3.85·26-s + 27-s + 6.38·29-s + 1.38·30-s + 10.5·31-s + 5.61·32-s + 2.23·33-s + ⋯ |
| L(s) = 1 | + 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.999·5-s + 0.252·6-s − 0.790·8-s + 0.333·9-s + 0.437·10-s + 0.674·11-s − 0.467·12-s + 1.72·13-s + 0.577·15-s + 0.463·16-s − 0.449·17-s + 0.145·18-s − 0.708·19-s − 0.809·20-s + 0.294·22-s − 0.962·23-s − 0.456·24-s + 0.755·26-s + 0.192·27-s + 1.18·29-s + 0.252·30-s + 1.89·31-s + 0.993·32-s + 0.389·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.617181479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617181479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 0.145T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.94T + 71T^{2} \) |
| 73 | \( 1 + 0.854T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 - 1.61T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−8.135018439475707136148569988074, −6.76163448756393245173886025389, −6.29732025880020580153408031847, −5.81292915114095099830026623942, −4.87677970282889573947362294086, −4.14079228678678275849108231307, −3.65656278811474393312339990066, −2.72844873640226872150572682207, −1.80445012584167116594210500994, −0.892951721473480871712300133031,
0.892951721473480871712300133031, 1.80445012584167116594210500994, 2.72844873640226872150572682207, 3.65656278811474393312339990066, 4.14079228678678275849108231307, 4.87677970282889573947362294086, 5.81292915114095099830026623942, 6.29732025880020580153408031847, 6.76163448756393245173886025389, 8.135018439475707136148569988074
