| L(s) = 1 | − 2.31·2-s − 3-s + 3.36·4-s − 2.86·5-s + 2.31·6-s + 3.60·7-s − 3.15·8-s + 9-s + 6.63·10-s + 0.489·11-s − 3.36·12-s + 2.88·13-s − 8.34·14-s + 2.86·15-s + 0.587·16-s − 6.07·17-s − 2.31·18-s − 0.208·19-s − 9.64·20-s − 3.60·21-s − 1.13·22-s − 23-s + 3.15·24-s + 3.21·25-s − 6.69·26-s − 27-s + 12.1·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.577·3-s + 1.68·4-s − 1.28·5-s + 0.945·6-s + 1.36·7-s − 1.11·8-s + 0.333·9-s + 2.09·10-s + 0.147·11-s − 0.971·12-s + 0.801·13-s − 2.22·14-s + 0.740·15-s + 0.146·16-s − 1.47·17-s − 0.545·18-s − 0.0479·19-s − 2.15·20-s − 0.785·21-s − 0.241·22-s − 0.208·23-s + 0.644·24-s + 0.643·25-s − 1.31·26-s − 0.192·27-s + 2.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 0.489T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 + 6.07T + 17T^{2} \) |
| 19 | \( 1 + 0.208T + 19T^{2} \) |
| 31 | \( 1 + 6.93T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 - 7.52T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 + 4.65T + 73T^{2} \) |
| 79 | \( 1 - 7.85T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 5.18T + 97T^{2} \) |
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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
−8.750652651743011110796509969674, −8.053057333773210839742654322788, −7.47172161203717727814945993965, −6.85579159464019066907054011228, −5.76460145640624722984801268172, −4.56360807886342905455191418759, −3.92428242165628635517154250027, −2.21528660966155935518241439739, −1.19932364405041157630126009864, 0,
1.19932364405041157630126009864, 2.21528660966155935518241439739, 3.92428242165628635517154250027, 4.56360807886342905455191418759, 5.76460145640624722984801268172, 6.85579159464019066907054011228, 7.47172161203717727814945993965, 8.053057333773210839742654322788, 8.750652651743011110796509969674
