L(s) = 1 | − 1.65·2-s − 1.94·3-s + 0.754·4-s − 5-s + 3.22·6-s + 7-s + 2.06·8-s + 0.767·9-s + 1.65·10-s − 1.86·11-s − 1.46·12-s + 5.05·13-s − 1.65·14-s + 1.94·15-s − 4.93·16-s + 5.41·17-s − 1.27·18-s − 0.656·19-s − 0.754·20-s − 1.94·21-s + 3.09·22-s + 1.52·23-s − 4.01·24-s + 25-s − 8.38·26-s + 4.33·27-s + 0.754·28-s + ⋯ |
L(s) = 1 | − 1.17·2-s − 1.12·3-s + 0.377·4-s − 0.447·5-s + 1.31·6-s + 0.377·7-s + 0.730·8-s + 0.255·9-s + 0.524·10-s − 0.563·11-s − 0.422·12-s + 1.40·13-s − 0.443·14-s + 0.501·15-s − 1.23·16-s + 1.31·17-s − 0.300·18-s − 0.150·19-s − 0.168·20-s − 0.423·21-s + 0.660·22-s + 0.317·23-s − 0.819·24-s + 0.200·25-s − 1.64·26-s + 0.833·27-s + 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 + 1.94T + 3T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 0.656T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 - 2.97T + 29T^{2} \) |
| 31 | \( 1 + 0.866T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 7.99T + 43T^{2} \) |
| 47 | \( 1 + 8.07T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 9.44T + 59T^{2} \) |
| 61 | \( 1 + 6.35T + 61T^{2} \) |
| 67 | \( 1 - 8.23T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 5.83T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 3.77T + 83T^{2} \) |
| 89 | \( 1 + 3.71T + 89T^{2} \) |
| 97 | \( 1 - 8.86T + 97T^{2} \) |
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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
−7.75636078223576973473413663823, −6.79170504501392662395565244245, −6.31712518256733044910545384456, −5.21144999104295693072816639464, −5.05663687271232037756003659223, −3.91677204878791519728905997398, −3.10497836776422249291794415366, −1.66139100154074774566185745688, −0.962553622715792141076987139273, 0,
0.962553622715792141076987139273, 1.66139100154074774566185745688, 3.10497836776422249291794415366, 3.91677204878791519728905997398, 5.05663687271232037756003659223, 5.21144999104295693072816639464, 6.31712518256733044910545384456, 6.79170504501392662395565244245, 7.75636078223576973473413663823