L(s) = 1 | − 2-s + 2.03·3-s + 4-s − 0.214·5-s − 2.03·6-s − 0.150·7-s − 8-s + 1.13·9-s + 0.214·10-s + 5.83·11-s + 2.03·12-s + 3.19·13-s + 0.150·14-s − 0.437·15-s + 16-s + 5.22·17-s − 1.13·18-s + 7.03·19-s − 0.214·20-s − 0.305·21-s − 5.83·22-s − 4.03·23-s − 2.03·24-s − 4.95·25-s − 3.19·26-s − 3.78·27-s − 0.150·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.17·3-s + 0.5·4-s − 0.0961·5-s − 0.830·6-s − 0.0567·7-s − 0.353·8-s + 0.379·9-s + 0.0679·10-s + 1.75·11-s + 0.587·12-s + 0.884·13-s + 0.0400·14-s − 0.112·15-s + 0.250·16-s + 1.26·17-s − 0.268·18-s + 1.61·19-s − 0.0480·20-s − 0.0665·21-s − 1.24·22-s − 0.842·23-s − 0.415·24-s − 0.990·25-s − 0.625·26-s − 0.728·27-s − 0.0283·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.576669714\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.576669714\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 - 2.03T + 3T^{2} \) |
| 5 | \( 1 + 0.214T + 5T^{2} \) |
| 7 | \( 1 + 0.150T + 7T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 7.03T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 - 9.36T + 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 3.97T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 0.989T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 9.44T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.313261525931303020168806722555, −8.030309483337243609150472436304, −7.25936308976131921302779552198, −6.32683941344326456469607607737, −5.78099270250512688782979247678, −4.39582625183640805758959998290, −3.44139752954949844481162825204, −3.14236572990232284642572992497, −1.77424754917288316421144416705, −1.07189681868536935263443415951,
1.07189681868536935263443415951, 1.77424754917288316421144416705, 3.14236572990232284642572992497, 3.44139752954949844481162825204, 4.39582625183640805758959998290, 5.78099270250512688782979247678, 6.32683941344326456469607607737, 7.25936308976131921302779552198, 8.030309483337243609150472436304, 8.313261525931303020168806722555