| L(s) = 1 | + 2.37·5-s + 4.37·7-s + 11-s − 0.372·13-s − 5.37·17-s + 0.627·19-s − 0.372·23-s + 0.627·25-s − 4.37·29-s + 6.37·31-s + 10.3·35-s + 8.74·37-s − 11.7·41-s − 1.74·43-s + 4.37·47-s + 12.1·49-s + 0.744·53-s + 2.37·55-s + 7·59-s + 2.37·61-s − 0.883·65-s − 3.74·67-s − 4·71-s − 12.1·73-s + 4.37·77-s − 6.37·79-s + 9.62·83-s + ⋯ |
| L(s) = 1 | + 1.06·5-s + 1.65·7-s + 0.301·11-s − 0.103·13-s − 1.30·17-s + 0.144·19-s − 0.0776·23-s + 0.125·25-s − 0.811·29-s + 1.14·31-s + 1.75·35-s + 1.43·37-s − 1.83·41-s − 0.266·43-s + 0.637·47-s + 1.73·49-s + 0.102·53-s + 0.319·55-s + 0.911·59-s + 0.303·61-s − 0.109·65-s − 0.457·67-s − 0.474·71-s − 1.41·73-s + 0.498·77-s − 0.716·79-s + 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.061630358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061630358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 + 0.372T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 6.37T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 1.74T + 43T^{2} \) |
| 47 | \( 1 - 4.37T + 47T^{2} \) |
| 53 | \( 1 - 0.744T + 53T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 + 3.74T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 6.37T + 79T^{2} \) |
| 83 | \( 1 - 9.62T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−10.58578571309101544899384529323, −9.693888308866421029156850687573, −8.799904979851025540572241785058, −8.061958899966239295067836908081, −6.98208663927452500695709114485, −5.97116825900122923094083502182, −5.05650003332228128234271788280, −4.22163329341672737002141606332, −2.42441613490697030993622988545, −1.51854295860303128436601914260,
1.51854295860303128436601914260, 2.42441613490697030993622988545, 4.22163329341672737002141606332, 5.05650003332228128234271788280, 5.97116825900122923094083502182, 6.98208663927452500695709114485, 8.061958899966239295067836908081, 8.799904979851025540572241785058, 9.693888308866421029156850687573, 10.58578571309101544899384529323
