| L(s) = 1 | + 1.02e3·2-s − 1.14e5·3-s + 1.04e6·4-s − 9.76e6·5-s − 1.17e8·6-s − 2.03e8·7-s + 1.07e9·8-s + 2.72e9·9-s − 1.00e10·10-s − 4.40e10·11-s − 1.20e11·12-s − 1.80e11·13-s − 2.08e11·14-s + 1.12e12·15-s + 1.09e12·16-s + 1.29e13·17-s + 2.79e12·18-s + 3.50e13·19-s − 1.02e13·20-s + 2.34e13·21-s − 4.50e13·22-s − 1.00e14·23-s − 1.23e14·24-s + 9.53e13·25-s − 1.85e14·26-s + 8.87e14·27-s − 2.13e14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.12·3-s + 0.5·4-s − 0.447·5-s − 0.794·6-s − 0.272·7-s + 0.353·8-s + 0.260·9-s − 0.316·10-s − 0.511·11-s − 0.561·12-s − 0.363·13-s − 0.192·14-s + 0.502·15-s + 0.250·16-s + 1.55·17-s + 0.184·18-s + 1.31·19-s − 0.223·20-s + 0.306·21-s − 0.361·22-s − 0.505·23-s − 0.397·24-s + 0.199·25-s − 0.257·26-s + 0.829·27-s − 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.681758740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681758740\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.02e3T \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 3 | \( 1 + 1.14e5T + 1.04e10T^{2} \) |
| 7 | \( 1 + 2.03e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 4.40e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 1.80e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.29e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.50e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.00e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.65e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 1.12e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.62e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 2.80e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.48e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 9.51e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.03e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.03e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 8.22e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.74e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.67e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.60e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 4.33e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.59e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 5.09e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.58e20T + 5.27e41T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.80904391050943419199518934072, −14.16960218827182429796380241887, −12.44505953192757271287888041570, −11.61423277013174624547472955210, −10.16505969652227915066826270464, −7.63250874819259180382847280239, −6.01437335541878318939133969791, −4.89552429151551962596653638970, −3.10198697336722944316667572319, −0.794842427459680101765561788939,
0.794842427459680101765561788939, 3.10198697336722944316667572319, 4.89552429151551962596653638970, 6.01437335541878318939133969791, 7.63250874819259180382847280239, 10.16505969652227915066826270464, 11.61423277013174624547472955210, 12.44505953192757271287888041570, 14.16960218827182429796380241887, 15.80904391050943419199518934072
