| L(s) = 1 | + 3.84·5-s − 0.680·7-s + 11-s + 13-s + 2.08·17-s + 6.37·19-s + 1.76·23-s + 9.82·25-s − 0.530·29-s + 2.32·31-s − 2.61·35-s − 2.44·37-s − 2.68·41-s + 4.08·43-s − 0.918·47-s − 6.53·49-s − 7.06·53-s + 3.84·55-s − 4.58·59-s − 4.91·61-s + 3.84·65-s + 10.4·67-s + 13.9·71-s − 10.0·73-s − 0.680·77-s + 7.61·79-s + 2.47·83-s + ⋯ |
| L(s) = 1 | + 1.72·5-s − 0.257·7-s + 0.301·11-s + 0.277·13-s + 0.506·17-s + 1.46·19-s + 0.367·23-s + 1.96·25-s − 0.0984·29-s + 0.417·31-s − 0.442·35-s − 0.401·37-s − 0.418·41-s + 0.623·43-s − 0.134·47-s − 0.933·49-s − 0.969·53-s + 0.519·55-s − 0.596·59-s − 0.629·61-s + 0.477·65-s + 1.27·67-s + 1.65·71-s − 1.17·73-s − 0.0775·77-s + 0.856·79-s + 0.271·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.120959206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.120959206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 + 0.680T + 7T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 + 0.530T + 29T^{2} \) |
| 31 | \( 1 - 2.32T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 + 0.918T + 47T^{2} \) |
| 53 | \( 1 + 7.06T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 + 4.91T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 2.44T + 97T^{2} \) |
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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
−8.301662660461276773805480099054, −7.39787341383978756969971876896, −6.62227479272090526774972075502, −6.05084028867725130611765049763, −5.39034688960201825863557122085, −4.79427909239240187955894534557, −3.50119505670028830586712175531, −2.83658050299034561555408332539, −1.81037982602525731152633603216, −1.04247602499522219836499116111,
1.04247602499522219836499116111, 1.81037982602525731152633603216, 2.83658050299034561555408332539, 3.50119505670028830586712175531, 4.79427909239240187955894534557, 5.39034688960201825863557122085, 6.05084028867725130611765049763, 6.62227479272090526774972075502, 7.39787341383978756969971876896, 8.301662660461276773805480099054
