L(s) = 1 | − 2.41·2-s + 2.41·3-s + 3.82·4-s − 5-s − 5.82·6-s − 2.82·7-s − 4.41·8-s + 2.82·9-s + 2.41·10-s − 0.414·11-s + 9.24·12-s − 3.82·13-s + 6.82·14-s − 2.41·15-s + 2.99·16-s + 0.828·17-s − 6.82·18-s + 6·19-s − 3.82·20-s − 6.82·21-s + 0.999·22-s + 3.65·23-s − 10.6·24-s − 4·25-s + 9.24·26-s − 0.414·27-s − 10.8·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.39·3-s + 1.91·4-s − 0.447·5-s − 2.37·6-s − 1.06·7-s − 1.56·8-s + 0.942·9-s + 0.763·10-s − 0.124·11-s + 2.66·12-s − 1.06·13-s + 1.82·14-s − 0.623·15-s + 0.749·16-s + 0.200·17-s − 1.60·18-s + 1.37·19-s − 0.856·20-s − 1.49·21-s + 0.213·22-s + 0.762·23-s − 2.17·24-s − 0.800·25-s + 1.81·26-s − 0.0797·27-s − 2.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4520178616\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4520178616\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−17.31343986241515149330655261806, −16.05888388043654018981880321944, −15.23224358305790267581443761926, −13.66732976667500941952922868953, −11.93782397100020205133590566736, −10.04252502469782990793395660236, −9.347008880241771359683944351195, −8.111694777874299421477288589426, −7.11407190598914430481785081554, −2.92477068592233051723034627885,
2.92477068592233051723034627885, 7.11407190598914430481785081554, 8.111694777874299421477288589426, 9.347008880241771359683944351195, 10.04252502469782990793395660236, 11.93782397100020205133590566736, 13.66732976667500941952922868953, 15.23224358305790267581443761926, 16.05888388043654018981880321944, 17.31343986241515149330655261806