L(s) = 1 | + 2-s + 3-s + 4-s − 0.266·5-s + 6-s + 0.661·7-s + 8-s + 9-s − 0.266·10-s + 6.00·11-s + 12-s + 4.92·13-s + 0.661·14-s − 0.266·15-s + 16-s − 17-s + 18-s + 2.93·19-s − 0.266·20-s + 0.661·21-s + 6.00·22-s + 3.23·23-s + 24-s − 4.92·25-s + 4.92·26-s + 27-s + 0.661·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.119·5-s + 0.408·6-s + 0.249·7-s + 0.353·8-s + 0.333·9-s − 0.0843·10-s + 1.81·11-s + 0.288·12-s + 1.36·13-s + 0.176·14-s − 0.0689·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.673·19-s − 0.0596·20-s + 0.144·21-s + 1.28·22-s + 0.675·23-s + 0.204·24-s − 0.985·25-s + 0.965·26-s + 0.192·27-s + 0.124·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.145500060\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.145500060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 0.266T + 5T^{2} \) |
| 7 | \( 1 - 0.661T + 7T^{2} \) |
| 11 | \( 1 - 6.00T + 11T^{2} \) |
| 13 | \( 1 - 4.92T + 13T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 - 7.24T + 43T^{2} \) |
| 47 | \( 1 + 7.06T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 - 9.08T + 67T^{2} \) |
| 71 | \( 1 - 0.403T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 + 3.21T + 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 - 9.29T + 89T^{2} \) |
| 97 | \( 1 + 9.48T + 97T^{2} \) |
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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
−8.110511649280919022850901135053, −7.26719810172754407210846294426, −6.50674767295582904490498543844, −6.13130693211787784550384823264, −5.05401721014930468928972435714, −4.29327286451243797158440685970, −3.61592553540299387490775940999, −3.12853154319458095474387536186, −1.79301271320798307341508595704, −1.19942123326799244765080431792,
1.19942123326799244765080431792, 1.79301271320798307341508595704, 3.12853154319458095474387536186, 3.61592553540299387490775940999, 4.29327286451243797158440685970, 5.05401721014930468928972435714, 6.13130693211787784550384823264, 6.50674767295582904490498543844, 7.26719810172754407210846294426, 8.110511649280919022850901135053