L(s) = 1 | + 0.655·3-s − 0.415·7-s − 2.56·9-s − 11-s − 4·13-s + 6.51·17-s − 5.20·19-s − 0.272·21-s + 8.54·23-s − 3.65·27-s + 0.895·29-s + 6.73·31-s − 0.655·33-s − 8.96·37-s − 2.62·39-s + 10.0·41-s − 4.78·43-s + 5.61·47-s − 6.82·49-s + 4.27·51-s − 10.0·53-s − 3.41·57-s + 1.63·59-s + 7.10·61-s + 1.06·63-s + 10.6·67-s + 5.60·69-s + ⋯ |
L(s) = 1 | + 0.378·3-s − 0.157·7-s − 0.856·9-s − 0.301·11-s − 1.10·13-s + 1.58·17-s − 1.19·19-s − 0.0595·21-s + 1.78·23-s − 0.702·27-s + 0.166·29-s + 1.21·31-s − 0.114·33-s − 1.47·37-s − 0.420·39-s + 1.57·41-s − 0.730·43-s + 0.819·47-s − 0.975·49-s + 0.598·51-s − 1.37·53-s − 0.452·57-s + 0.212·59-s + 0.909·61-s + 0.134·63-s + 1.30·67-s + 0.674·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.736552075\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.736552075\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 0.655T + 3T^{2} \) |
| 7 | \( 1 + 0.415T + 7T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 - 8.54T + 23T^{2} \) |
| 29 | \( 1 - 0.895T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 + 8.96T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.78T + 43T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 - 7.10T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 9.56T + 89T^{2} \) |
| 97 | \( 1 + 0.591T + 97T^{2} \) |
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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
−8.267713399926790079305086735256, −7.79283292081148369702718620134, −6.94365750365857115993155114788, −6.21198807321180116731212770728, −5.26377794650394002440929528523, −4.81509378480335830567588789132, −3.56675081128709938327826192532, −2.92439155020918833656467107960, −2.15838103760475043972081169758, −0.70543311350786148130358088540,
0.70543311350786148130358088540, 2.15838103760475043972081169758, 2.92439155020918833656467107960, 3.56675081128709938327826192532, 4.81509378480335830567588789132, 5.26377794650394002440929528523, 6.21198807321180116731212770728, 6.94365750365857115993155114788, 7.79283292081148369702718620134, 8.267713399926790079305086735256