L(s) = 1 | − 2.27·2-s − 3-s + 3.19·4-s − 3.13·5-s + 2.27·6-s − 0.0802·7-s − 2.71·8-s + 9-s + 7.13·10-s − 5.00·11-s − 3.19·12-s + 3.99·13-s + 0.182·14-s + 3.13·15-s − 0.193·16-s − 17-s − 2.27·18-s + 2.46·19-s − 9.99·20-s + 0.0802·21-s + 11.3·22-s + 2.52·23-s + 2.71·24-s + 4.80·25-s − 9.09·26-s − 27-s − 0.256·28-s + ⋯ |
L(s) = 1 | − 1.61·2-s − 0.577·3-s + 1.59·4-s − 1.40·5-s + 0.930·6-s − 0.0303·7-s − 0.960·8-s + 0.333·9-s + 2.25·10-s − 1.50·11-s − 0.921·12-s + 1.10·13-s + 0.0488·14-s + 0.808·15-s − 0.0484·16-s − 0.242·17-s − 0.537·18-s + 0.564·19-s − 2.23·20-s + 0.0175·21-s + 2.42·22-s + 0.526·23-s + 0.554·24-s + 0.961·25-s − 1.78·26-s − 0.192·27-s − 0.0483·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2186563991\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2186563991\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 5 | \( 1 + 3.13T + 5T^{2} \) |
| 7 | \( 1 + 0.0802T + 7T^{2} \) |
| 11 | \( 1 + 5.00T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 + 0.805T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 0.183T + 41T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 - 4.80T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 8.96T + 67T^{2} \) |
| 71 | \( 1 - 7.92T + 71T^{2} \) |
| 73 | \( 1 + 4.57T + 73T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 0.860T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.221451270047633436100498586670, −7.905132222777442750194439981850, −7.33424278429207122055980717223, −6.58391063212021201441686635092, −5.61203779038204681156806583752, −4.71039145256412016775294770572, −3.73469929812338811735114973710, −2.77063807001297044599733374754, −1.46845257088923813657293036187, −0.36704975522914059910474212238,
0.36704975522914059910474212238, 1.46845257088923813657293036187, 2.77063807001297044599733374754, 3.73469929812338811735114973710, 4.71039145256412016775294770572, 5.61203779038204681156806583752, 6.58391063212021201441686635092, 7.33424278429207122055980717223, 7.905132222777442750194439981850, 8.221451270047633436100498586670