| L(s) = 1 | + 2.40·2-s − 0.650·3-s + 3.78·4-s + 5-s − 1.56·6-s − 3.90·7-s + 4.29·8-s − 2.57·9-s + 2.40·10-s − 1.72·11-s − 2.46·12-s − 6.24·13-s − 9.40·14-s − 0.650·15-s + 2.76·16-s − 3.09·17-s − 6.19·18-s + 5.07·19-s + 3.78·20-s + 2.54·21-s − 4.14·22-s + 0.896·23-s − 2.79·24-s + 25-s − 15.0·26-s + 3.62·27-s − 14.8·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.375·3-s + 1.89·4-s + 0.447·5-s − 0.639·6-s − 1.47·7-s + 1.51·8-s − 0.858·9-s + 0.760·10-s − 0.519·11-s − 0.711·12-s − 1.73·13-s − 2.51·14-s − 0.168·15-s + 0.691·16-s − 0.750·17-s − 1.46·18-s + 1.16·19-s + 0.846·20-s + 0.555·21-s − 0.883·22-s + 0.186·23-s − 0.571·24-s + 0.200·25-s − 2.94·26-s + 0.698·27-s − 2.79·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 + 0.650T + 3T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 - 0.896T + 23T^{2} \) |
| 29 | \( 1 + 6.36T + 29T^{2} \) |
| 31 | \( 1 - 0.222T + 31T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 - 0.225T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.918661149074562115526601006009, −7.47848405880115337507161457364, −6.85470956204465371997098399369, −6.14817346420050230459155915174, −5.38249662677822658319859348525, −4.98909852435261538188499222008, −3.75805717907820238914370363620, −2.87976710806083267400187814984, −2.37776644526143718226799363277, 0,
2.37776644526143718226799363277, 2.87976710806083267400187814984, 3.75805717907820238914370363620, 4.98909852435261538188499222008, 5.38249662677822658319859348525, 6.14817346420050230459155915174, 6.85470956204465371997098399369, 7.47848405880115337507161457364, 8.918661149074562115526601006009
