| L(s) = 1 | − 0.714·3-s − 1.61·7-s − 2.48·9-s + 1.28·11-s + 13-s − 1.33·17-s − 0.950·19-s + 1.15·21-s + 5.58·23-s + 3.92·27-s + 0.538·29-s − 4.65·31-s − 0.918·33-s − 7.99·37-s − 0.714·39-s + 5.35·41-s + 3.44·43-s − 5.60·47-s − 4.38·49-s + 0.952·51-s − 0.657·53-s + 0.679·57-s + 5.78·59-s − 11.4·61-s + 4.02·63-s + 4.42·67-s − 3.99·69-s + ⋯ |
| L(s) = 1 | − 0.412·3-s − 0.611·7-s − 0.829·9-s + 0.387·11-s + 0.277·13-s − 0.323·17-s − 0.218·19-s + 0.252·21-s + 1.16·23-s + 0.755·27-s + 0.0999·29-s − 0.836·31-s − 0.159·33-s − 1.31·37-s − 0.114·39-s + 0.836·41-s + 0.524·43-s − 0.818·47-s − 0.625·49-s + 0.133·51-s − 0.0903·53-s + 0.0900·57-s + 0.753·59-s − 1.46·61-s + 0.507·63-s + 0.541·67-s − 0.480·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092495818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092495818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.714T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 + 0.950T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 0.538T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 + 7.99T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 + 5.60T + 47T^{2} \) |
| 53 | \( 1 + 0.657T + 53T^{2} \) |
| 59 | \( 1 - 5.78T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 4.42T + 67T^{2} \) |
| 71 | \( 1 + 2.20T + 71T^{2} \) |
| 73 | \( 1 - 3.94T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 0.0463T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−7.60432353088323369233980769368, −6.71150674959455936134246648711, −6.45204084124099650338357933516, −5.57889218642189364558088700800, −5.08052553658568078277934513098, −4.15028805715623971357946027901, −3.35523591183526115835151931049, −2.73135798987667201298475166725, −1.64490005824690639537897358954, −0.50235418887516635924699512734,
0.50235418887516635924699512734, 1.64490005824690639537897358954, 2.73135798987667201298475166725, 3.35523591183526115835151931049, 4.15028805715623971357946027901, 5.08052553658568078277934513098, 5.57889218642189364558088700800, 6.45204084124099650338357933516, 6.71150674959455936134246648711, 7.60432353088323369233980769368
