L(s) = 1 | − 3.04·2-s + 3·3-s + 1.27·4-s − 15.0·5-s − 9.13·6-s + 20.4·8-s + 9·9-s + 45.8·10-s − 11·11-s + 3.82·12-s − 29.9·13-s − 45.1·15-s − 72.5·16-s + 48.0·17-s − 27.4·18-s − 86.0·19-s − 19.2·20-s + 33.5·22-s + 96.2·23-s + 61.4·24-s + 101.·25-s + 91.1·26-s + 27·27-s + 61.9·29-s + 137.·30-s + 159.·31-s + 57.2·32-s + ⋯ |
L(s) = 1 | − 1.07·2-s + 0.577·3-s + 0.159·4-s − 1.34·5-s − 0.621·6-s + 0.905·8-s + 0.333·9-s + 1.44·10-s − 0.301·11-s + 0.0920·12-s − 0.638·13-s − 0.777·15-s − 1.13·16-s + 0.685·17-s − 0.358·18-s − 1.03·19-s − 0.214·20-s + 0.324·22-s + 0.872·23-s + 0.522·24-s + 0.812·25-s + 0.687·26-s + 0.192·27-s + 0.396·29-s + 0.837·30-s + 0.922·31-s + 0.315·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 + 3.04T + 8T^{2} \) |
| 5 | \( 1 + 15.0T + 125T^{2} \) |
| 13 | \( 1 + 29.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 61.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 26.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 341.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 80.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 109.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 645.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 505.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 264.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 865.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 402.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 171.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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
−8.424967107529447348128256289767, −8.125052735310865744321201326083, −7.37062106689972471829993938668, −6.72129264849081914929039058850, −5.09376868872700467665722238823, −4.34837358635790966435412459114, −3.46186493901307059938297741051, −2.33512230506469217832493818583, −0.978893324969226094585264949181, 0,
0.978893324969226094585264949181, 2.33512230506469217832493818583, 3.46186493901307059938297741051, 4.34837358635790966435412459114, 5.09376868872700467665722238823, 6.72129264849081914929039058850, 7.37062106689972471829993938668, 8.125052735310865744321201326083, 8.424967107529447348128256289767