L(s) = 1 | − 2-s − 3-s + 4-s − 0.976·5-s + 6-s − 3.87·7-s − 8-s + 9-s + 0.976·10-s − 0.0200·11-s − 12-s + 4.09·13-s + 3.87·14-s + 0.976·15-s + 16-s + 17-s − 18-s + 4.76·19-s − 0.976·20-s + 3.87·21-s + 0.0200·22-s − 0.204·23-s + 24-s − 4.04·25-s − 4.09·26-s − 27-s − 3.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.436·5-s + 0.408·6-s − 1.46·7-s − 0.353·8-s + 0.333·9-s + 0.308·10-s − 0.00604·11-s − 0.288·12-s + 1.13·13-s + 1.03·14-s + 0.252·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.09·19-s − 0.218·20-s + 0.845·21-s + 0.00427·22-s − 0.0427·23-s + 0.204·24-s − 0.809·25-s − 0.802·26-s − 0.192·27-s − 0.732·28-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
\(L(1)\) |
\(\approx\) |
\(0.6295080439\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6295080439\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.976T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 + 0.0200T + 11T^{2} \) |
| 13 | \( 1 - 4.09T + 13T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 + 0.204T + 23T^{2} \) |
| 29 | \( 1 + 6.91T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 + 0.0763T + 37T^{2} \) |
| 41 | \( 1 - 1.72T + 41T^{2} \) |
| 43 | \( 1 - 0.671T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 3.54T + 53T^{2} \) |
| 61 | \( 1 + 6.17T + 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 + 9.33T + 79T^{2} \) |
| 83 | \( 1 + 2.90T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 6.56T + 97T^{2} \) |
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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
−7.986572448891959049597526508313, −7.35349068042489106447335530336, −6.76918865287018913626956323407, −5.84867673825276004372228727395, −5.71590823052884104930080160361, −4.25970985768730243327192617664, −3.55493179362930580647966548304, −2.87842662427460912157650665548, −1.53186868896680664083990763998, −0.49285769817402343732726080945,
0.49285769817402343732726080945, 1.53186868896680664083990763998, 2.87842662427460912157650665548, 3.55493179362930580647966548304, 4.25970985768730243327192617664, 5.71590823052884104930080160361, 5.84867673825276004372228727395, 6.76918865287018913626956323407, 7.35349068042489106447335530336, 7.986572448891959049597526508313