| L(s) = 1 | + 0.239·2-s − 1.94·4-s − 1.18·5-s − 7-s − 0.942·8-s − 0.282·10-s + 3.70·11-s + 13-s − 0.239·14-s + 3.66·16-s + 6.94·17-s + 1.94·19-s + 2.29·20-s + 0.885·22-s + 5.60·23-s − 3.60·25-s + 0.239·26-s + 1.94·28-s − 0.239·29-s + 1.66·31-s + 2.76·32-s + 1.66·34-s + 1.18·35-s − 9.54·37-s + 0.464·38-s + 1.11·40-s + 10.1·41-s + ⋯ |
| L(s) = 1 | + 0.169·2-s − 0.971·4-s − 0.528·5-s − 0.377·7-s − 0.333·8-s − 0.0893·10-s + 1.11·11-s + 0.277·13-s − 0.0639·14-s + 0.915·16-s + 1.68·17-s + 0.445·19-s + 0.513·20-s + 0.188·22-s + 1.16·23-s − 0.720·25-s + 0.0468·26-s + 0.367·28-s − 0.0444·29-s + 0.298·31-s + 0.488·32-s + 0.284·34-s + 0.199·35-s − 1.56·37-s + 0.0753·38-s + 0.176·40-s + 1.59·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.195794606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195794606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 0.239T + 2T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 6.94T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 - 5.60T + 23T^{2} \) |
| 29 | \( 1 + 0.239T + 29T^{2} \) |
| 31 | \( 1 - 1.66T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−10.69008582043613053035910427525, −9.667202982911172467936908893282, −9.114854047520996330393765466281, −8.145175021850109391241477118141, −7.24642224008905757908686137464, −6.04365020176797562726267992751, −5.11473368907394368245575790124, −3.92990874286156576178906274739, −3.30014270255967393831870514018, −1.01347993391273402801437244725,
1.01347993391273402801437244725, 3.30014270255967393831870514018, 3.92990874286156576178906274739, 5.11473368907394368245575790124, 6.04365020176797562726267992751, 7.24642224008905757908686137464, 8.145175021850109391241477118141, 9.114854047520996330393765466281, 9.667202982911172467936908893282, 10.69008582043613053035910427525
