| L(s) = 1 | − 2-s − 2.67·3-s + 4-s + 2.43·5-s + 2.67·6-s − 4.11·7-s − 8-s + 4.16·9-s − 2.43·10-s + 1.05·11-s − 2.67·12-s − 4.86·13-s + 4.11·14-s − 6.51·15-s + 16-s − 5.22·17-s − 4.16·18-s − 4.63·19-s + 2.43·20-s + 11.0·21-s − 1.05·22-s − 7.57·23-s + 2.67·24-s + 0.921·25-s + 4.86·26-s − 3.13·27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.54·3-s + 0.5·4-s + 1.08·5-s + 1.09·6-s − 1.55·7-s − 0.353·8-s + 1.38·9-s − 0.769·10-s + 0.319·11-s − 0.772·12-s − 1.34·13-s + 1.09·14-s − 1.68·15-s + 0.250·16-s − 1.26·17-s − 0.982·18-s − 1.06·19-s + 0.544·20-s + 2.40·21-s − 0.225·22-s − 1.58·23-s + 0.546·24-s + 0.184·25-s + 0.953·26-s − 0.602·27-s − 0.776·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1529182217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1529182217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.67T + 3T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 + 7.57T + 23T^{2} \) |
| 29 | \( 1 - 0.794T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 + 0.955T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 61 | \( 1 - 5.19T + 61T^{2} \) |
| 67 | \( 1 + 8.04T + 67T^{2} \) |
| 71 | \( 1 + 7.05T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 - 2.83T + 79T^{2} \) |
| 83 | \( 1 + 9.91T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.714674550774569809838647767348, −7.40137191049073438460028976787, −6.78217105745366514586005268691, −6.08921911485100687352355759955, −6.00312117654552200661621795224, −4.89776516342145727693439544026, −3.98437862277849930569458638500, −2.57978862571714357693287654322, −1.84686809710605821117412078719, −0.25276371303031164446199104738,
0.25276371303031164446199104738, 1.84686809710605821117412078719, 2.57978862571714357693287654322, 3.98437862277849930569458638500, 4.89776516342145727693439544026, 6.00312117654552200661621795224, 6.08921911485100687352355759955, 6.78217105745366514586005268691, 7.40137191049073438460028976787, 8.714674550774569809838647767348
