| L(s) = 1 | + 2.48·2-s + 3-s + 4.18·4-s + 2.95·5-s + 2.48·6-s + 5.44·8-s + 9-s + 7.34·10-s − 0.300·11-s + 4.18·12-s + 3.25·13-s + 2.95·15-s + 5.15·16-s − 2.43·17-s + 2.48·18-s − 0.970·19-s + 12.3·20-s − 0.747·22-s + 23-s + 5.44·24-s + 3.71·25-s + 8.09·26-s + 27-s − 6.80·29-s + 7.34·30-s + 2.23·31-s + 1.94·32-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.577·3-s + 2.09·4-s + 1.32·5-s + 1.01·6-s + 1.92·8-s + 0.333·9-s + 2.32·10-s − 0.0905·11-s + 1.20·12-s + 0.902·13-s + 0.762·15-s + 1.28·16-s − 0.590·17-s + 0.586·18-s − 0.222·19-s + 2.76·20-s − 0.159·22-s + 0.208·23-s + 1.11·24-s + 0.743·25-s + 1.58·26-s + 0.192·27-s − 1.26·29-s + 1.34·30-s + 0.401·31-s + 0.344·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.629363623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.629363623\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 5 | \( 1 - 2.95T + 5T^{2} \) |
| 11 | \( 1 + 0.300T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 + 0.970T + 19T^{2} \) |
| 29 | \( 1 + 6.80T + 29T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 0.769T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 + 4.88T + 47T^{2} \) |
| 53 | \( 1 + 3.72T + 53T^{2} \) |
| 59 | \( 1 - 8.22T + 59T^{2} \) |
| 61 | \( 1 + 7.13T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 + 1.85T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 - 7.19T + 89T^{2} \) |
| 97 | \( 1 - 1.43T + 97T^{2} \) |
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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
−8.662406855887012634566000541926, −7.63276394811647019415696252839, −6.60067076873590689480051236824, −6.34917617213041277924953551551, −5.40160407686665775430047073552, −4.91530246823671478622653869388, −3.83413937820517047447996653156, −3.25363805786163228216390445372, −2.20704363425822251950301597255, −1.69181330322467521533750203194,
1.69181330322467521533750203194, 2.20704363425822251950301597255, 3.25363805786163228216390445372, 3.83413937820517047447996653156, 4.91530246823671478622653869388, 5.40160407686665775430047073552, 6.34917617213041277924953551551, 6.60067076873590689480051236824, 7.63276394811647019415696252839, 8.662406855887012634566000541926
