L(s) = 1 | + 2-s + 4-s − 5-s − 4.73·7-s + 8-s − 10-s + 5.46·11-s − 5.46·13-s − 4.73·14-s + 16-s − 5.46·17-s + 6.19·19-s − 20-s + 5.46·22-s + 8·23-s + 25-s − 5.46·26-s − 4.73·28-s − 4.92·29-s + 0.732·31-s + 32-s − 5.46·34-s + 4.73·35-s + 37-s + 6.19·38-s − 40-s + 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.78·7-s + 0.353·8-s − 0.316·10-s + 1.64·11-s − 1.51·13-s − 1.26·14-s + 0.250·16-s − 1.32·17-s + 1.42·19-s − 0.223·20-s + 1.16·22-s + 1.66·23-s + 0.200·25-s − 1.07·26-s − 0.894·28-s − 0.915·29-s + 0.131·31-s + 0.176·32-s − 0.937·34-s + 0.799·35-s + 0.164·37-s + 1.00·38-s − 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.107713504\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.107713504\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 5.46T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 0.732T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 3.66T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.988876113344153921594680461266, −7.36680006553795715284092477221, −7.09758896522786767159942168519, −6.47428863621165422036753705211, −5.62869059140768206230099576192, −4.66297590856814571721982725487, −3.88201525601083615060169804748, −3.19349904571341647880365677377, −2.39797329385675176809751330251, −0.76184886513914383464049980410,
0.76184886513914383464049980410, 2.39797329385675176809751330251, 3.19349904571341647880365677377, 3.88201525601083615060169804748, 4.66297590856814571721982725487, 5.62869059140768206230099576192, 6.47428863621165422036753705211, 7.09758896522786767159942168519, 7.36680006553795715284092477221, 8.988876113344153921594680461266