L(s) = 1 | + 2.77·2-s + 5.68·4-s + 5-s + 2.27·7-s + 10.2·8-s + 2.77·10-s + 0.435·13-s + 6.31·14-s + 16.9·16-s + 5·17-s − 4.69·19-s + 5.68·20-s + 0.845·23-s + 25-s + 1.20·26-s + 12.9·28-s − 2.65·29-s − 4.66·31-s + 26.5·32-s + 13.8·34-s + 2.27·35-s − 8.86·37-s − 13.0·38-s + 10.2·40-s − 4.29·41-s − 7.00·43-s + 2.34·46-s + ⋯ |
L(s) = 1 | + 1.96·2-s + 2.84·4-s + 0.447·5-s + 0.860·7-s + 3.61·8-s + 0.876·10-s + 0.120·13-s + 1.68·14-s + 4.23·16-s + 1.21·17-s − 1.07·19-s + 1.27·20-s + 0.176·23-s + 0.200·25-s + 0.236·26-s + 2.44·28-s − 0.493·29-s − 0.838·31-s + 4.69·32-s + 2.37·34-s + 0.384·35-s − 1.45·37-s − 2.11·38-s + 1.61·40-s − 0.670·41-s − 1.06·43-s + 0.345·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.537491701\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.537491701\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 13 | \( 1 - 0.435T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 - 0.845T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 8.86T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 + 7.00T + 43T^{2} \) |
| 47 | \( 1 + 0.468T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 + 2.96T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 8.42T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 8.64T + 97T^{2} \) |
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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
−7.908416487811939050235394140111, −7.11420144121401011161779336758, −6.56552911859353385764556423307, −5.63296800132723038215738805621, −5.32458139376413375476231775165, −4.57960268277726161201212268983, −3.77400040160597159949692558295, −3.10858709295713051244626503155, −2.04722676270099946303300742936, −1.50895330436667793693632952535,
1.50895330436667793693632952535, 2.04722676270099946303300742936, 3.10858709295713051244626503155, 3.77400040160597159949692558295, 4.57960268277726161201212268983, 5.32458139376413375476231775165, 5.63296800132723038215738805621, 6.56552911859353385764556423307, 7.11420144121401011161779336758, 7.908416487811939050235394140111