L(s) = 1 | + 1.43·2-s + 1.66·3-s + 0.0643·4-s + 5-s + 2.39·6-s − 2.09·7-s − 2.78·8-s − 0.227·9-s + 1.43·10-s + 2.86·11-s + 0.107·12-s − 6.31·13-s − 3.00·14-s + 1.66·15-s − 4.12·16-s + 17-s − 0.327·18-s + 5.28·19-s + 0.0643·20-s − 3.48·21-s + 4.11·22-s + 8.05·23-s − 4.63·24-s + 25-s − 9.07·26-s − 5.37·27-s − 0.134·28-s + ⋯ |
L(s) = 1 | + 1.01·2-s + 0.961·3-s + 0.0321·4-s + 0.447·5-s + 0.976·6-s − 0.791·7-s − 0.983·8-s − 0.0758·9-s + 0.454·10-s + 0.863·11-s + 0.0309·12-s − 1.75·13-s − 0.803·14-s + 0.429·15-s − 1.03·16-s + 0.242·17-s − 0.0771·18-s + 1.21·19-s + 0.0143·20-s − 0.760·21-s + 0.877·22-s + 1.67·23-s − 0.945·24-s + 0.200·25-s − 1.77·26-s − 1.03·27-s − 0.0254·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 3 | \( 1 - 1.66T + 3T^{2} \) |
| 7 | \( 1 + 2.09T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 - 8.05T + 23T^{2} \) |
| 29 | \( 1 + 5.17T + 29T^{2} \) |
| 31 | \( 1 + 0.868T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 + 2.96T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 + 3.45T + 67T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 + 1.60T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.55567453950844280938564136485, −6.94456879340962515150813730146, −6.26487132706429711604393765246, −5.20174929600051873300492527383, −5.05208120470007565322959756315, −3.84151131041809759989738352416, −3.13736866845384857251098106754, −2.86523864989938552164637087485, −1.68623207423224200234128698215, 0,
1.68623207423224200234128698215, 2.86523864989938552164637087485, 3.13736866845384857251098106754, 3.84151131041809759989738352416, 5.05208120470007565322959756315, 5.20174929600051873300492527383, 6.26487132706429711604393765246, 6.94456879340962515150813730146, 7.55567453950844280938564136485