| L(s) = 1 | − 2-s − 1.78·3-s + 4-s − 3.15·5-s + 1.78·6-s − 2.13·7-s − 8-s + 0.180·9-s + 3.15·10-s − 1.22·11-s − 1.78·12-s + 1.64·13-s + 2.13·14-s + 5.61·15-s + 16-s − 2.01·17-s − 0.180·18-s + 19-s − 3.15·20-s + 3.79·21-s + 1.22·22-s + 0.392·23-s + 1.78·24-s + 4.92·25-s − 1.64·26-s + 5.02·27-s − 2.13·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.02·3-s + 0.5·4-s − 1.40·5-s + 0.728·6-s − 0.805·7-s − 0.353·8-s + 0.0601·9-s + 0.996·10-s − 0.370·11-s − 0.514·12-s + 0.455·13-s + 0.569·14-s + 1.45·15-s + 0.250·16-s − 0.487·17-s − 0.0425·18-s + 0.229·19-s − 0.704·20-s + 0.829·21-s + 0.261·22-s + 0.0819·23-s + 0.364·24-s + 0.985·25-s − 0.321·26-s + 0.967·27-s − 0.402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02392859988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02392859988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 + 2.01T + 17T^{2} \) |
| 23 | \( 1 - 0.392T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + 8.13T + 31T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 + 1.75T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 8.70T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 + 5.07T + 61T^{2} \) |
| 67 | \( 1 + 8.40T + 67T^{2} \) |
| 71 | \( 1 - 0.457T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.66376656968011501306767581728, −7.24898807550907095404075730136, −6.53200233493180236401652934507, −5.90584847779374799482465681654, −5.14775172567845830687977106920, −4.26681341565213476250794254105, −3.46286606510516810492535159684, −2.78368032542112024452562739586, −1.36186045403112138687294878352, −0.094758640171002227031269808487,
0.094758640171002227031269808487, 1.36186045403112138687294878352, 2.78368032542112024452562739586, 3.46286606510516810492535159684, 4.26681341565213476250794254105, 5.14775172567845830687977106920, 5.90584847779374799482465681654, 6.53200233493180236401652934507, 7.24898807550907095404075730136, 7.66376656968011501306767581728
