| L(s) = 1 | + 2-s + 0.115·3-s + 4-s − 3.86·5-s + 0.115·6-s + 2.67·7-s + 8-s − 2.98·9-s − 3.86·10-s − 1.98·11-s + 0.115·12-s + 5.84·13-s + 2.67·14-s − 0.446·15-s + 16-s − 5.78·17-s − 2.98·18-s + 2.76·19-s − 3.86·20-s + 0.309·21-s − 1.98·22-s + 5.37·23-s + 0.115·24-s + 9.97·25-s + 5.84·26-s − 0.691·27-s + 2.67·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0666·3-s + 0.5·4-s − 1.73·5-s + 0.0471·6-s + 1.01·7-s + 0.353·8-s − 0.995·9-s − 1.22·10-s − 0.599·11-s + 0.0333·12-s + 1.62·13-s + 0.715·14-s − 0.115·15-s + 0.250·16-s − 1.40·17-s − 0.703·18-s + 0.634·19-s − 0.865·20-s + 0.0674·21-s − 0.424·22-s + 1.12·23-s + 0.0235·24-s + 1.99·25-s + 1.14·26-s − 0.133·27-s + 0.506·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.115T + 3T^{2} \) |
| 5 | \( 1 + 3.86T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 + 7.08T + 31T^{2} \) |
| 37 | \( 1 - 4.16T + 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 + 2.22T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 9.95T + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 - 7.56T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.082802166521403441457244214473, −7.41709278851368183615519886535, −6.68108639764881792923698893437, −5.62983522751665346177510334480, −5.00899917475198564738746721240, −4.18356675120490245586076369920, −3.55810601912136961235262532795, −2.80837497545676517913439228257, −1.48914689750280577752504262654, 0,
1.48914689750280577752504262654, 2.80837497545676517913439228257, 3.55810601912136961235262532795, 4.18356675120490245586076369920, 5.00899917475198564738746721240, 5.62983522751665346177510334480, 6.68108639764881792923698893437, 7.41709278851368183615519886535, 8.082802166521403441457244214473
