| L(s) = 1 | + 0.347·2-s − 1.87·4-s + 0.652·5-s + 0.532·7-s − 1.34·8-s + 0.226·10-s − 11-s − 2.30·13-s + 0.184·14-s + 3.29·16-s − 4.41·17-s − 4.18·19-s − 1.22·20-s − 0.347·22-s + 1.41·23-s − 4.57·25-s − 0.800·26-s − 28-s + 6.35·29-s − 2.16·31-s + 3.83·32-s − 1.53·34-s + 0.347·35-s − 3.16·37-s − 1.45·38-s − 0.879·40-s − 9.31·41-s + ⋯ |
| L(s) = 1 | + 0.245·2-s − 0.939·4-s + 0.291·5-s + 0.201·7-s − 0.476·8-s + 0.0716·10-s − 0.301·11-s − 0.639·13-s + 0.0493·14-s + 0.822·16-s − 1.06·17-s − 0.960·19-s − 0.274·20-s − 0.0740·22-s + 0.294·23-s − 0.914·25-s − 0.157·26-s − 0.188·28-s + 1.18·29-s − 0.388·31-s + 0.678·32-s − 0.262·34-s + 0.0587·35-s − 0.519·37-s − 0.235·38-s − 0.139·40-s − 1.45·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 5 | \( 1 - 0.652T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 + 2.16T + 31T^{2} \) |
| 37 | \( 1 + 3.16T + 37T^{2} \) |
| 41 | \( 1 + 9.31T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 3.86T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 - 9.98T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−9.813528084012545849001413704947, −8.715451400683312969934453774565, −8.322185163147674644761020573079, −7.06579729254783944256956272214, −6.14106593976144323564404634477, −5.05887063621192910141549236423, −4.50617322891500742157460201944, −3.31908492930294815809901345461, −1.97679894025303810451404739737, 0,
1.97679894025303810451404739737, 3.31908492930294815809901345461, 4.50617322891500742157460201944, 5.05887063621192910141549236423, 6.14106593976144323564404634477, 7.06579729254783944256956272214, 8.322185163147674644761020573079, 8.715451400683312969934453774565, 9.813528084012545849001413704947
