| L(s) = 1 | − 1.91·2-s − 2.56·3-s + 1.65·4-s − 2·5-s + 4.91·6-s + 0.656·8-s + 3.59·9-s + 3.82·10-s − 2·11-s − 4.25·12-s − 2.34·13-s + 5.13·15-s − 4.56·16-s + 3.22·17-s − 6.88·18-s − 2.08·19-s − 3.31·20-s + 3.82·22-s − 6.56·23-s − 1.68·24-s − 25-s + 4.48·26-s − 1.53·27-s + 7.82·29-s − 9.82·30-s − 3.82·31-s + 7.42·32-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 1.48·3-s + 0.828·4-s − 0.894·5-s + 2.00·6-s + 0.232·8-s + 1.19·9-s + 1.20·10-s − 0.603·11-s − 1.22·12-s − 0.649·13-s + 1.32·15-s − 1.14·16-s + 0.782·17-s − 1.62·18-s − 0.478·19-s − 0.740·20-s + 0.815·22-s − 1.36·23-s − 0.344·24-s − 0.200·25-s + 0.878·26-s − 0.296·27-s + 1.45·29-s − 1.79·30-s − 0.686·31-s + 1.31·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 - 4.28T + 37T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 4.51T + 61T^{2} \) |
| 67 | \( 1 - 6.51T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 8.45T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 - 9.64T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.706909176348304945242291817216, −7.83867745020134996769628499545, −7.50746772565269887436584758510, −6.54211987417189412491932952262, −5.70284724374799198204905315437, −4.77813060699874174911354764206, −4.00427073390273818881907577066, −2.36230765223698651342811300968, −0.882468315462357779645774155532, 0,
0.882468315462357779645774155532, 2.36230765223698651342811300968, 4.00427073390273818881907577066, 4.77813060699874174911354764206, 5.70284724374799198204905315437, 6.54211987417189412491932952262, 7.50746772565269887436584758510, 7.83867745020134996769628499545, 8.706909176348304945242291817216
