| L(s) = 1 | − 2.16·2-s − 3-s + 2.66·4-s − 0.855·5-s + 2.16·6-s + 2.67·7-s − 1.44·8-s + 9-s + 1.84·10-s + 2.50·11-s − 2.66·12-s + 1.50·13-s − 5.77·14-s + 0.855·15-s − 2.22·16-s + 0.755·17-s − 2.16·18-s − 1.29·19-s − 2.28·20-s − 2.67·21-s − 5.40·22-s + 0.0548·23-s + 1.44·24-s − 4.26·25-s − 3.26·26-s − 27-s + 7.12·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.577·3-s + 1.33·4-s − 0.382·5-s + 0.881·6-s + 1.01·7-s − 0.509·8-s + 0.333·9-s + 0.584·10-s + 0.753·11-s − 0.769·12-s + 0.418·13-s − 1.54·14-s + 0.220·15-s − 0.555·16-s + 0.183·17-s − 0.509·18-s − 0.298·19-s − 0.510·20-s − 0.583·21-s − 1.15·22-s + 0.0114·23-s + 0.294·24-s − 0.853·25-s − 0.639·26-s − 0.192·27-s + 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6812187515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6812187515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 2671 | \( 1 - T \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 5 | \( 1 + 0.855T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 - 0.755T + 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 - 0.0548T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 + 3.18T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 - 0.451T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 4.93T + 97T^{2} \) |
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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
−7.967285908952748445658055018272, −7.34612447119217403016378110561, −6.72503345335959866834623429428, −5.94009478684395412017946521142, −5.11565411799267974867309450426, −4.27616429413756485664328345420, −3.56442325934937449621273853452, −2.06643411102696721018681217729, −1.53759286294149673253362856066, −0.56353575486102573807356145004,
0.56353575486102573807356145004, 1.53759286294149673253362856066, 2.06643411102696721018681217729, 3.56442325934937449621273853452, 4.27616429413756485664328345420, 5.11565411799267974867309450426, 5.94009478684395412017946521142, 6.72503345335959866834623429428, 7.34612447119217403016378110561, 7.967285908952748445658055018272
