| L(s) = 1 | − 2.82·3-s + 1.41·5-s + 5.00·9-s + 4·11-s + 4.24·13-s − 4.00·15-s + 1.41·17-s + 2.82·19-s − 4·23-s − 2.99·25-s − 5.65·27-s + 8·29-s − 11.3·33-s − 8·37-s − 12·39-s − 7.07·41-s − 4·43-s + 7.07·45-s + 5.65·47-s − 4.00·51-s + 10·53-s + 5.65·55-s − 8.00·57-s + 14.1·59-s − 7.07·61-s + 6·65-s + 11.3·69-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.632·5-s + 1.66·9-s + 1.20·11-s + 1.17·13-s − 1.03·15-s + 0.342·17-s + 0.648·19-s − 0.834·23-s − 0.599·25-s − 1.08·27-s + 1.48·29-s − 1.96·33-s − 1.31·37-s − 1.92·39-s − 1.10·41-s − 0.609·43-s + 1.05·45-s + 0.825·47-s − 0.560·51-s + 1.37·53-s + 0.762·55-s − 1.05·57-s + 1.84·59-s − 0.905·61-s + 0.744·65-s + 1.36·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8723641270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8723641270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 1.41T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.01801209257579565799880886231, −11.78118085089026838605485024886, −10.58045995448057079474808428178, −9.877393439712082231162545028013, −8.587224786167342026200073187279, −6.89650934972302960219554185576, −6.14539973457519346514706405452, −5.33135340261650426661592219336, −3.91584139815081451345714498381, −1.32286566317686927886391412472,
1.32286566317686927886391412472, 3.91584139815081451345714498381, 5.33135340261650426661592219336, 6.14539973457519346514706405452, 6.89650934972302960219554185576, 8.587224786167342026200073187279, 9.877393439712082231162545028013, 10.58045995448057079474808428178, 11.78118085089026838605485024886, 12.01801209257579565799880886231
