| L(s) = 1 | + 2-s + 0.232·3-s + 4-s − 1.02·5-s + 0.232·6-s − 0.709·7-s + 8-s − 2.94·9-s − 1.02·10-s + 3.09·11-s + 0.232·12-s − 3.68·13-s − 0.709·14-s − 0.239·15-s + 16-s + 4.38·17-s − 2.94·18-s − 4.12·19-s − 1.02·20-s − 0.165·21-s + 3.09·22-s + 8.56·23-s + 0.232·24-s − 3.94·25-s − 3.68·26-s − 1.38·27-s − 0.709·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.134·3-s + 0.5·4-s − 0.459·5-s + 0.0950·6-s − 0.268·7-s + 0.353·8-s − 0.981·9-s − 0.324·10-s + 0.932·11-s + 0.0672·12-s − 1.02·13-s − 0.189·14-s − 0.0617·15-s + 0.250·16-s + 1.06·17-s − 0.694·18-s − 0.945·19-s − 0.229·20-s − 0.0360·21-s + 0.659·22-s + 1.78·23-s + 0.0475·24-s − 0.789·25-s − 0.722·26-s − 0.266·27-s − 0.134·28-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 | \( 1 - 0.232T + 3T^{2} \) |
| 5 | \( 1 + 1.02T + 5T^{2} \) |
| 7 | \( 1 + 0.709T + 7T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + 4.12T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 + 9.20T + 29T^{2} \) |
| 31 | \( 1 - 0.0180T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 3.04T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 - 1.74T + 47T^{2} \) |
| 53 | \( 1 + 2.32T + 53T^{2} \) |
| 59 | \( 1 + 3.33T + 59T^{2} \) |
| 61 | \( 1 + 4.58T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 4.91T + 79T^{2} \) |
| 83 | \( 1 + 5.06T + 83T^{2} \) |
| 89 | \( 1 + 3.08T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.922403069419232785159824220578, −7.28163133985379964519022059347, −6.60484962352624079604729357401, −5.70925584079667957408009554559, −5.15207490657727482734513316513, −4.15617847296681357314455524749, −3.44655494715071163399780089271, −2.75424445533397199879930471783, −1.61813489436439727220359739961, 0,
1.61813489436439727220359739961, 2.75424445533397199879930471783, 3.44655494715071163399780089271, 4.15617847296681357314455524749, 5.15207490657727482734513316513, 5.70925584079667957408009554559, 6.60484962352624079604729357401, 7.28163133985379964519022059347, 7.922403069419232785159824220578
