L(s) = 1 | − 0.573·2-s + 0.0256·3-s − 1.67·4-s − 5-s − 0.0147·6-s − 7-s + 2.10·8-s − 2.99·9-s + 0.573·10-s − 2.70·11-s − 0.0428·12-s + 5.99·13-s + 0.573·14-s − 0.0256·15-s + 2.13·16-s + 8.13·17-s + 1.72·18-s + 1.85·19-s + 1.67·20-s − 0.0256·21-s + 1.55·22-s − 4.26·23-s + 0.0539·24-s + 25-s − 3.43·26-s − 0.153·27-s + 1.67·28-s + ⋯ |
L(s) = 1 | − 0.405·2-s + 0.0147·3-s − 0.835·4-s − 0.447·5-s − 0.00600·6-s − 0.377·7-s + 0.744·8-s − 0.999·9-s + 0.181·10-s − 0.815·11-s − 0.0123·12-s + 1.66·13-s + 0.153·14-s − 0.00661·15-s + 0.533·16-s + 1.97·17-s + 0.405·18-s + 0.425·19-s + 0.373·20-s − 0.00559·21-s + 0.330·22-s − 0.888·23-s + 0.0110·24-s + 0.200·25-s − 0.674·26-s − 0.0295·27-s + 0.315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9127290071\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9127290071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 0.573T + 2T^{2} \) |
| 3 | \( 1 - 0.0256T + 3T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 - 5.99T + 13T^{2} \) |
| 17 | \( 1 - 8.13T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 + 4.26T + 23T^{2} \) |
| 29 | \( 1 - 0.728T + 29T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 + 5.42T + 47T^{2} \) |
| 53 | \( 1 - 0.229T + 53T^{2} \) |
| 59 | \( 1 - 0.971T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 8.03T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 7.41T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.79655226320629798119672173685, −7.65920676369978930938110640769, −6.22147942434394174499910003399, −5.80285427253990283582181944903, −5.12972768210335579688706698135, −4.19357581250372998704353358853, −3.41418574089104616794711940371, −2.97014512204313021464487411007, −1.44130949092240449466397059938, −0.54653945494915157278587393863,
0.54653945494915157278587393863, 1.44130949092240449466397059938, 2.97014512204313021464487411007, 3.41418574089104616794711940371, 4.19357581250372998704353358853, 5.12972768210335579688706698135, 5.80285427253990283582181944903, 6.22147942434394174499910003399, 7.65920676369978930938110640769, 7.79655226320629798119672173685