| L(s) = 1 | + 2-s − 3.19·3-s + 4-s + 3.31·5-s − 3.19·6-s − 7-s + 8-s + 7.22·9-s + 3.31·10-s + 1.11·11-s − 3.19·12-s + 3.66·13-s − 14-s − 10.6·15-s + 16-s − 0.834·17-s + 7.22·18-s + 6.33·19-s + 3.31·20-s + 3.19·21-s + 1.11·22-s + 5.24·23-s − 3.19·24-s + 5.99·25-s + 3.66·26-s − 13.5·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.84·3-s + 0.5·4-s + 1.48·5-s − 1.30·6-s − 0.377·7-s + 0.353·8-s + 2.40·9-s + 1.04·10-s + 0.335·11-s − 0.923·12-s + 1.01·13-s − 0.267·14-s − 2.73·15-s + 0.250·16-s − 0.202·17-s + 1.70·18-s + 1.45·19-s + 0.741·20-s + 0.697·21-s + 0.237·22-s + 1.09·23-s − 0.652·24-s + 1.19·25-s + 0.718·26-s − 2.60·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.734922449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.734922449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 11 | \( 1 - 1.11T + 11T^{2} \) |
| 13 | \( 1 - 3.66T + 13T^{2} \) |
| 17 | \( 1 + 0.834T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 - 4.75T + 37T^{2} \) |
| 41 | \( 1 - 6.06T + 41T^{2} \) |
| 43 | \( 1 + 9.13T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6.14T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 0.111T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 1.82T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 6.51T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.73653358534064362853357669857, −6.79034368671403040826430600510, −6.43280917466138082987891041679, −5.90993019397480468724952689620, −5.31835225277323302838014367994, −4.83671414753110296530767807605, −3.85153952562575859935815650970, −2.83082874083188817738357162337, −1.56155429638325402296406189276, −0.964176211355997831143008537420,
0.964176211355997831143008537420, 1.56155429638325402296406189276, 2.83082874083188817738357162337, 3.85153952562575859935815650970, 4.83671414753110296530767807605, 5.31835225277323302838014367994, 5.90993019397480468724952689620, 6.43280917466138082987891041679, 6.79034368671403040826430600510, 7.73653358534064362853357669857
