L(s) = 1 | − 2.58·2-s + 0.518·3-s + 4.70·4-s − 1.61·5-s − 1.34·6-s − 6.99·8-s − 2.73·9-s + 4.17·10-s − 2.70·11-s + 2.43·12-s − 0.835·15-s + 8.69·16-s + 3.12·17-s + 7.06·18-s − 3.68·19-s − 7.57·20-s + 7.00·22-s + 1.98·23-s − 3.62·24-s − 2.40·25-s − 2.97·27-s − 5.37·29-s + 2.16·30-s − 10.4·31-s − 8.52·32-s − 1.40·33-s − 8.09·34-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 0.299·3-s + 2.35·4-s − 0.720·5-s − 0.547·6-s − 2.47·8-s − 0.910·9-s + 1.31·10-s − 0.815·11-s + 0.703·12-s − 0.215·15-s + 2.17·16-s + 0.758·17-s + 1.66·18-s − 0.844·19-s − 1.69·20-s + 1.49·22-s + 0.414·23-s − 0.739·24-s − 0.480·25-s − 0.571·27-s − 0.997·29-s + 0.395·30-s − 1.88·31-s − 1.50·32-s − 0.244·33-s − 1.38·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1731049108\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1731049108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 - 0.518T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 5.95T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 + 1.05T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 + 9.10T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.942493569419467607503492126385, −7.47658246876982959765440135750, −6.86955792579682191923987600881, −5.85847774930058255310557529316, −5.35161708243319025969926647649, −3.96368787745338902001418008846, −3.18496171918670629447811939978, −2.41027330176394323973381715320, −1.60690712512778993744994400324, −0.25744442300102052837029991019,
0.25744442300102052837029991019, 1.60690712512778993744994400324, 2.41027330176394323973381715320, 3.18496171918670629447811939978, 3.96368787745338902001418008846, 5.35161708243319025969926647649, 5.85847774930058255310557529316, 6.86955792579682191923987600881, 7.47658246876982959765440135750, 7.942493569419467607503492126385