| L(s) = 1 | − 1.56·2-s − 0.440·3-s + 0.444·4-s − 3.20·5-s + 0.688·6-s − 3.01·7-s + 2.43·8-s − 2.80·9-s + 5.01·10-s + 11-s − 0.195·12-s − 1.07·13-s + 4.71·14-s + 1.41·15-s − 4.69·16-s + 17-s + 4.38·18-s + 1.71·19-s − 1.42·20-s + 1.32·21-s − 1.56·22-s − 3.83·23-s − 1.07·24-s + 5.29·25-s + 1.68·26-s + 2.55·27-s − 1.34·28-s + ⋯ |
| L(s) = 1 | − 1.10·2-s − 0.254·3-s + 0.222·4-s − 1.43·5-s + 0.281·6-s − 1.14·7-s + 0.859·8-s − 0.935·9-s + 1.58·10-s + 0.301·11-s − 0.0565·12-s − 0.299·13-s + 1.26·14-s + 0.364·15-s − 1.17·16-s + 0.242·17-s + 1.03·18-s + 0.392·19-s − 0.318·20-s + 0.289·21-s − 0.333·22-s − 0.800·23-s − 0.218·24-s + 1.05·25-s + 0.330·26-s + 0.492·27-s − 0.253·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07107028467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07107028467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 3 | \( 1 + 0.440T + 3T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 7 | \( 1 + 3.01T + 7T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 - 4.93T + 29T^{2} \) |
| 31 | \( 1 - 9.73T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 9.99T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 0.595T + 79T^{2} \) |
| 83 | \( 1 - 1.27T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.034457179806552516204338931425, −7.36809762751931735328269569847, −6.59144927843376830053319958879, −6.04112021607679399182402497706, −4.86030603565640142810471782515, −4.31815630869005742995554622410, −3.34084851709470877609714691071, −2.82606989622782797045546409735, −1.30297047508339007103422844942, −0.17294195134779789383398485120,
0.17294195134779789383398485120, 1.30297047508339007103422844942, 2.82606989622782797045546409735, 3.34084851709470877609714691071, 4.31815630869005742995554622410, 4.86030603565640142810471782515, 6.04112021607679399182402497706, 6.59144927843376830053319958879, 7.36809762751931735328269569847, 8.034457179806552516204338931425
