| L(s) = 1 | + 0.369·2-s − 1.27·3-s − 1.86·4-s − 0.472·6-s + 1.49·7-s − 1.42·8-s − 1.36·9-s + 3.68·11-s + 2.37·12-s + 3.61·13-s + 0.552·14-s + 3.19·16-s − 5.00·17-s − 0.506·18-s − 6.14·19-s − 1.90·21-s + 1.36·22-s + 1.06·23-s + 1.82·24-s + 1.33·26-s + 5.57·27-s − 2.78·28-s + 0.426·29-s − 10.0·31-s + 4.03·32-s − 4.70·33-s − 1.84·34-s + ⋯ |
| L(s) = 1 | + 0.261·2-s − 0.737·3-s − 0.931·4-s − 0.192·6-s + 0.564·7-s − 0.504·8-s − 0.456·9-s + 1.11·11-s + 0.686·12-s + 1.00·13-s + 0.147·14-s + 0.799·16-s − 1.21·17-s − 0.119·18-s − 1.40·19-s − 0.416·21-s + 0.290·22-s + 0.221·23-s + 0.372·24-s + 0.262·26-s + 1.07·27-s − 0.525·28-s + 0.0792·29-s − 1.81·31-s + 0.713·32-s − 0.819·33-s − 0.317·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 0.369T + 2T^{2} \) |
| 3 | \( 1 + 1.27T + 3T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 19 | \( 1 + 6.14T + 19T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 - 0.426T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 8.49T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 + 8.00T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 - 1.62T + 71T^{2} \) |
| 73 | \( 1 + 8.77T + 73T^{2} \) |
| 79 | \( 1 - 5.75T + 79T^{2} \) |
| 83 | \( 1 + 1.38T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−9.201494522786011758492437889933, −8.804195996087228114691402666230, −8.026238170011618857088958812206, −6.50601006398646804262922770885, −6.14602241258999256371964122774, −5.00758690350498657226946838259, −4.36349535666295560606446320470, −3.40759582315194519582462784292, −1.62474874899470402365962240706, 0,
1.62474874899470402365962240706, 3.40759582315194519582462784292, 4.36349535666295560606446320470, 5.00758690350498657226946838259, 6.14602241258999256371964122774, 6.50601006398646804262922770885, 8.026238170011618857088958812206, 8.804195996087228114691402666230, 9.201494522786011758492437889933
