L(s) = 1 | − 2-s + 1.94·3-s + 4-s + 4.05·5-s − 1.94·6-s − 2.97·7-s − 8-s + 0.781·9-s − 4.05·10-s − 0.141·11-s + 1.94·12-s + 1.83·13-s + 2.97·14-s + 7.88·15-s + 16-s + 17-s − 0.781·18-s + 5.45·19-s + 4.05·20-s − 5.78·21-s + 0.141·22-s − 5.75·23-s − 1.94·24-s + 11.4·25-s − 1.83·26-s − 4.31·27-s − 2.97·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.12·3-s + 0.5·4-s + 1.81·5-s − 0.793·6-s − 1.12·7-s − 0.353·8-s + 0.260·9-s − 1.28·10-s − 0.0426·11-s + 0.561·12-s + 0.507·13-s + 0.794·14-s + 2.03·15-s + 0.250·16-s + 0.242·17-s − 0.184·18-s + 1.25·19-s + 0.907·20-s − 1.26·21-s + 0.0301·22-s − 1.20·23-s − 0.396·24-s + 2.29·25-s − 0.359·26-s − 0.830·27-s − 0.561·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.416266290\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.416266290\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 - 4.05T + 5T^{2} \) |
| 7 | \( 1 + 2.97T + 7T^{2} \) |
| 11 | \( 1 + 0.141T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 29 | \( 1 - 8.77T + 29T^{2} \) |
| 31 | \( 1 + 8.77T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 - 9.04T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 61 | \( 1 - 4.66T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 7.08T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−9.186256325168965117674674795849, −8.720055909862057112747353087698, −7.71561003646703733187670048224, −6.89416140996220334923182343406, −5.93804235061051792013161688168, −5.62801124033845297349265682285, −3.87654867329511264764311949063, −2.82642180391026994423545833741, −2.37649042735818453360118670619, −1.16519217030859795622893970244,
1.16519217030859795622893970244, 2.37649042735818453360118670619, 2.82642180391026994423545833741, 3.87654867329511264764311949063, 5.62801124033845297349265682285, 5.93804235061051792013161688168, 6.89416140996220334923182343406, 7.71561003646703733187670048224, 8.720055909862057112747353087698, 9.186256325168965117674674795849