L(s) = 1 | + 1.34·2-s − 3-s − 0.181·4-s − 1.77·5-s − 1.34·6-s + 1.31·7-s − 2.94·8-s + 9-s − 2.39·10-s − 1.43·11-s + 0.181·12-s + 13-s + 1.77·14-s + 1.77·15-s − 3.60·16-s − 0.353·17-s + 1.34·18-s + 6.34·19-s + 0.322·20-s − 1.31·21-s − 1.93·22-s + 3.44·23-s + 2.94·24-s − 1.85·25-s + 1.34·26-s − 27-s − 0.238·28-s + ⋯ |
L(s) = 1 | + 0.953·2-s − 0.577·3-s − 0.0908·4-s − 0.792·5-s − 0.550·6-s + 0.496·7-s − 1.04·8-s + 0.333·9-s − 0.756·10-s − 0.433·11-s + 0.0524·12-s + 0.277·13-s + 0.473·14-s + 0.457·15-s − 0.900·16-s − 0.0857·17-s + 0.317·18-s + 1.45·19-s + 0.0720·20-s − 0.286·21-s − 0.413·22-s + 0.717·23-s + 0.600·24-s − 0.371·25-s + 0.264·26-s − 0.192·27-s − 0.0451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 17 | \( 1 + 0.353T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 3.44T + 23T^{2} \) |
| 29 | \( 1 - 8.26T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.305T + 41T^{2} \) |
| 43 | \( 1 + 6.02T + 43T^{2} \) |
| 47 | \( 1 + 9.42T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 8.84T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 6.44T + 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 9.14T + 83T^{2} \) |
| 89 | \( 1 + 9.98T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.117893060756582212313261364616, −7.18163754639743546721855329693, −6.49908240647177347467616083529, −5.57469296399471639484978834086, −4.95849618041190703340865727911, −4.50794382909415007331140427012, −3.51042696328199607627057525284, −2.91886916540551991001846413227, −1.31896700603841157736665563159, 0,
1.31896700603841157736665563159, 2.91886916540551991001846413227, 3.51042696328199607627057525284, 4.50794382909415007331140427012, 4.95849618041190703340865727911, 5.57469296399471639484978834086, 6.49908240647177347467616083529, 7.18163754639743546721855329693, 8.117893060756582212313261364616