L(s) = 1 | − 2.25·2-s − 2.77·3-s + 3.08·4-s + 0.288·5-s + 6.26·6-s + 1.78·7-s − 2.45·8-s + 4.70·9-s − 0.649·10-s − 11-s − 8.57·12-s + 5.07·13-s − 4.02·14-s − 0.799·15-s − 0.637·16-s − 2.51·17-s − 10.6·18-s − 2.44·19-s + 0.889·20-s − 4.95·21-s + 2.25·22-s − 7.23·23-s + 6.81·24-s − 4.91·25-s − 11.4·26-s − 4.72·27-s + 5.51·28-s + ⋯ |
L(s) = 1 | − 1.59·2-s − 1.60·3-s + 1.54·4-s + 0.128·5-s + 2.55·6-s + 0.675·7-s − 0.868·8-s + 1.56·9-s − 0.205·10-s − 0.301·11-s − 2.47·12-s + 1.40·13-s − 1.07·14-s − 0.206·15-s − 0.159·16-s − 0.610·17-s − 2.50·18-s − 0.561·19-s + 0.198·20-s − 1.08·21-s + 0.480·22-s − 1.50·23-s + 1.39·24-s − 0.983·25-s − 2.24·26-s − 0.910·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2606546118\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2606546118\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 3 | \( 1 + 2.77T + 3T^{2} \) |
| 5 | \( 1 - 0.288T + 5T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 7.23T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 4.64T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 0.0153T + 79T^{2} \) |
| 83 | \( 1 - 1.08T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−8.193964828623507730888358004857, −7.43632809048946002427684124281, −6.74766253229966475274855382628, −5.99206797568954720392662386084, −5.61667351589242879251602285157, −4.54775937043622716427900568152, −3.79631006313191177746729378364, −2.03020465977158906547472729774, −1.56082989382281963851004547209, −0.38343966772273819351003821141,
0.38343966772273819351003821141, 1.56082989382281963851004547209, 2.03020465977158906547472729774, 3.79631006313191177746729378364, 4.54775937043622716427900568152, 5.61667351589242879251602285157, 5.99206797568954720392662386084, 6.74766253229966475274855382628, 7.43632809048946002427684124281, 8.193964828623507730888358004857