L(s) = 1 | + 0.149·2-s − 3.21·3-s − 1.97·4-s + 1.07·5-s − 0.479·6-s − 3.58·7-s − 0.593·8-s + 7.36·9-s + 0.160·10-s + 11-s + 6.36·12-s − 0.534·14-s − 3.46·15-s + 3.86·16-s − 5.78·17-s + 1.09·18-s − 7.32·19-s − 2.12·20-s + 11.5·21-s + 0.149·22-s − 5.23·23-s + 1.90·24-s − 3.84·25-s − 14.0·27-s + 7.08·28-s − 3.62·29-s − 0.516·30-s + ⋯ |
L(s) = 1 | + 0.105·2-s − 1.85·3-s − 0.988·4-s + 0.481·5-s − 0.195·6-s − 1.35·7-s − 0.209·8-s + 2.45·9-s + 0.0507·10-s + 0.301·11-s + 1.83·12-s − 0.142·14-s − 0.894·15-s + 0.966·16-s − 1.40·17-s + 0.258·18-s − 1.68·19-s − 0.475·20-s + 2.51·21-s + 0.0317·22-s − 1.09·23-s + 0.389·24-s − 0.768·25-s − 2.70·27-s + 1.33·28-s − 0.673·29-s − 0.0942·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1825392501\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1825392501\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.149T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 + 7.32T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 + 3.91T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 0.145T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + 5.14T + 67T^{2} \) |
| 71 | \( 1 + 0.0702T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 + 3.82T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−9.380532215004770874427548122608, −8.752519495569772926127322960083, −7.34978606300870004730277960763, −6.37538881882222827142408888685, −6.14210670081655965146391385184, −5.35467192293402362481197082177, −4.32044256592303216070749664441, −3.86102902052414884581489944486, −1.98696540121566704007326017880, −0.30152771366293336387669340468,
0.30152771366293336387669340468, 1.98696540121566704007326017880, 3.86102902052414884581489944486, 4.32044256592303216070749664441, 5.35467192293402362481197082177, 6.14210670081655965146391385184, 6.37538881882222827142408888685, 7.34978606300870004730277960763, 8.752519495569772926127322960083, 9.380532215004770874427548122608