| L(s) = 1 | − 2.62·2-s + 1.59·3-s + 4.89·4-s + 2.83·5-s − 4.19·6-s − 4.08·7-s − 7.58·8-s − 0.452·9-s − 7.43·10-s − 4.35·11-s + 7.80·12-s + 1.46·13-s + 10.7·14-s + 4.52·15-s + 10.1·16-s + 0.159·17-s + 1.18·18-s + 6.75·19-s + 13.8·20-s − 6.51·21-s + 11.4·22-s + 2.68·23-s − 12.1·24-s + 3.02·25-s − 3.85·26-s − 5.51·27-s − 19.9·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.921·3-s + 2.44·4-s + 1.26·5-s − 1.71·6-s − 1.54·7-s − 2.68·8-s − 0.150·9-s − 2.35·10-s − 1.31·11-s + 2.25·12-s + 0.406·13-s + 2.86·14-s + 1.16·15-s + 2.53·16-s + 0.0386·17-s + 0.279·18-s + 1.54·19-s + 3.09·20-s − 1.42·21-s + 2.43·22-s + 0.560·23-s − 2.47·24-s + 0.605·25-s − 0.755·26-s − 1.06·27-s − 3.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6047 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 0.159T + 17T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 - 4.32T + 29T^{2} \) |
| 31 | \( 1 - 3.01T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 + 3.88T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 2.01T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 9.50T + 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 4.27T + 89T^{2} \) |
| 97 | \( 1 + 3.38T + 97T^{2} \) |
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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
−7.82194925313723267613131423762, −7.38143065269104393487254307589, −6.40097021812609786558860139298, −6.05043826917644178036101791862, −5.15996113800887414813084733009, −3.16317923077960602418712990987, −3.02199069289330077764558863714, −2.25673938977301365540055261128, −1.25084278180195073549858024395, 0,
1.25084278180195073549858024395, 2.25673938977301365540055261128, 3.02199069289330077764558863714, 3.16317923077960602418712990987, 5.15996113800887414813084733009, 6.05043826917644178036101791862, 6.40097021812609786558860139298, 7.38143065269104393487254307589, 7.82194925313723267613131423762
