| L(s) = 1 | − 2.16·2-s − 3-s + 2.70·4-s − 4.31·5-s + 2.16·6-s − 1.36·7-s − 1.52·8-s + 9-s + 9.34·10-s + 0.114·11-s − 2.70·12-s + 0.931·13-s + 2.95·14-s + 4.31·15-s − 2.10·16-s + 17-s − 2.16·18-s + 6.13·19-s − 11.6·20-s + 1.36·21-s − 0.248·22-s − 4.88·23-s + 1.52·24-s + 13.5·25-s − 2.02·26-s − 27-s − 3.68·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.577·3-s + 1.35·4-s − 1.92·5-s + 0.885·6-s − 0.515·7-s − 0.537·8-s + 0.333·9-s + 2.95·10-s + 0.0345·11-s − 0.779·12-s + 0.258·13-s + 0.790·14-s + 1.11·15-s − 0.526·16-s + 0.242·17-s − 0.511·18-s + 1.40·19-s − 2.60·20-s + 0.297·21-s − 0.0529·22-s − 1.01·23-s + 0.310·24-s + 2.71·25-s − 0.396·26-s − 0.192·27-s − 0.696·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3038390274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3038390274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 5 | \( 1 + 4.31T + 5T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 - 0.114T + 11T^{2} \) |
| 13 | \( 1 - 0.931T + 13T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 - 7.77T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 + 7.97T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 5.18T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 7.29T + 67T^{2} \) |
| 71 | \( 1 + 4.93T + 71T^{2} \) |
| 73 | \( 1 - 7.06T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.70961296322696988182505318467, −7.49174352191216062198675362160, −6.87318786050620703905798209871, −5.97208695842116909081033297249, −5.08072513745727184986481140748, −4.01956295626446807321274869511, −3.65168344128722728899315319988, −2.50576378693620560677401393959, −1.14822489971186625337981626595, −0.42785728165288075822887321304,
0.42785728165288075822887321304, 1.14822489971186625337981626595, 2.50576378693620560677401393959, 3.65168344128722728899315319988, 4.01956295626446807321274869511, 5.08072513745727184986481140748, 5.97208695842116909081033297249, 6.87318786050620703905798209871, 7.49174352191216062198675362160, 7.70961296322696988182505318467
