| L(s) = 1 | − 2.79·3-s − 2.34·5-s + 1.28·7-s + 4.80·9-s − 5.75·11-s + 0.304·13-s + 6.54·15-s + 4.18·17-s − 3.58·21-s + 6.47·23-s + 0.497·25-s − 5.04·27-s − 3.12·29-s − 6.44·31-s + 16.0·33-s − 3.01·35-s − 3.97·37-s − 0.850·39-s − 5.01·41-s + 0.989·43-s − 11.2·45-s + 4.39·47-s − 5.35·49-s − 11.6·51-s + 3.29·53-s + 13.4·55-s − 3.31·59-s + ⋯ |
| L(s) = 1 | − 1.61·3-s − 1.04·5-s + 0.485·7-s + 1.60·9-s − 1.73·11-s + 0.0843·13-s + 1.69·15-s + 1.01·17-s − 0.782·21-s + 1.35·23-s + 0.0994·25-s − 0.970·27-s − 0.580·29-s − 1.15·31-s + 2.79·33-s − 0.508·35-s − 0.654·37-s − 0.136·39-s − 0.783·41-s + 0.150·43-s − 1.67·45-s + 0.640·47-s − 0.764·49-s − 1.63·51-s + 0.452·53-s + 1.81·55-s − 0.431·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4052212559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4052212559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 - 0.304T + 13T^{2} \) |
| 17 | \( 1 - 4.18T + 17T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 - 0.989T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 + 3.31T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 + 8.57T + 79T^{2} \) |
| 83 | \( 1 - 9.76T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.79645067586470867576671987950, −7.47463099032890546450206296489, −6.74935693723740742674018029874, −5.67538666958053709755738235483, −5.28778564625223847454234990685, −4.77036987339765108509475465525, −3.82369921524329147139225918500, −2.92299090690959445868766258837, −1.52766214136784557851777538771, −0.37774433466473154522706771534,
0.37774433466473154522706771534, 1.52766214136784557851777538771, 2.92299090690959445868766258837, 3.82369921524329147139225918500, 4.77036987339765108509475465525, 5.28778564625223847454234990685, 5.67538666958053709755738235483, 6.74935693723740742674018029874, 7.47463099032890546450206296489, 7.79645067586470867576671987950
