| L(s) = 1 | + 0.543·2-s − 1.77·3-s − 1.70·4-s − 0.382·5-s − 0.966·6-s − 4.84·7-s − 2.01·8-s + 0.165·9-s − 0.207·10-s − 11-s + 3.03·12-s + 6.63·13-s − 2.63·14-s + 0.680·15-s + 2.31·16-s + 4.99·17-s + 0.0900·18-s + 7.01·19-s + 0.651·20-s + 8.62·21-s − 0.543·22-s − 7.26·23-s + 3.58·24-s − 4.85·25-s + 3.60·26-s + 5.04·27-s + 8.26·28-s + ⋯ |
| L(s) = 1 | + 0.384·2-s − 1.02·3-s − 0.852·4-s − 0.170·5-s − 0.394·6-s − 1.83·7-s − 0.711·8-s + 0.0552·9-s − 0.0656·10-s − 0.301·11-s + 0.875·12-s + 1.84·13-s − 0.703·14-s + 0.175·15-s + 0.579·16-s + 1.21·17-s + 0.0212·18-s + 1.60·19-s + 0.145·20-s + 1.88·21-s − 0.115·22-s − 1.51·23-s + 0.730·24-s − 0.970·25-s + 0.707·26-s + 0.970·27-s + 1.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6377593238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6377593238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.543T + 2T^{2} \) |
| 3 | \( 1 + 1.77T + 3T^{2} \) |
| 5 | \( 1 + 0.382T + 5T^{2} \) |
| 7 | \( 1 + 4.84T + 7T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 4.99T + 17T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 23 | \( 1 + 7.26T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 8.90T + 37T^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 + 0.438T + 43T^{2} \) |
| 47 | \( 1 + 9.41T + 47T^{2} \) |
| 53 | \( 1 - 8.67T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 67 | \( 1 - 6.48T + 67T^{2} \) |
| 71 | \( 1 + 3.31T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + 2.55T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−10.30348630182227295686737782478, −9.872008689382341729686668523326, −8.904731011071137054641638875185, −7.907844692232079902846152587134, −6.53953195457080325469799800182, −5.83755988916590472583747500677, −5.37561767375830250466571461489, −3.72692059035725046622670065535, −3.36531679350889306206748040489, −0.66728022197421492482165220902,
0.66728022197421492482165220902, 3.36531679350889306206748040489, 3.72692059035725046622670065535, 5.37561767375830250466571461489, 5.83755988916590472583747500677, 6.53953195457080325469799800182, 7.907844692232079902846152587134, 8.904731011071137054641638875185, 9.872008689382341729686668523326, 10.30348630182227295686737782478
