| L(s) = 1 | + 0.141·2-s − 1.97·4-s − 3.14·5-s − 4.94·7-s − 0.565·8-s − 0.446·10-s − 11-s − 5.08·13-s − 0.701·14-s + 3.87·16-s + 4.36·17-s + 3.23·19-s + 6.23·20-s − 0.141·22-s − 4.32·23-s + 4.91·25-s − 0.721·26-s + 9.78·28-s − 4.78·29-s + 3.82·31-s + 1.68·32-s + 0.620·34-s + 15.5·35-s + 3.16·37-s + 0.459·38-s + 1.77·40-s − 3.06·41-s + ⋯ |
| L(s) = 1 | + 0.100·2-s − 0.989·4-s − 1.40·5-s − 1.86·7-s − 0.199·8-s − 0.141·10-s − 0.301·11-s − 1.40·13-s − 0.187·14-s + 0.969·16-s + 1.05·17-s + 0.742·19-s + 1.39·20-s − 0.0302·22-s − 0.901·23-s + 0.982·25-s − 0.141·26-s + 1.84·28-s − 0.888·29-s + 0.686·31-s + 0.297·32-s + 0.106·34-s + 2.63·35-s + 0.520·37-s + 0.0745·38-s + 0.281·40-s − 0.478·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.141T + 2T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 7 | \( 1 + 4.94T + 7T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 4.32T + 23T^{2} \) |
| 29 | \( 1 + 4.78T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 + 3.06T + 41T^{2} \) |
| 43 | \( 1 + 8.71T + 43T^{2} \) |
| 47 | \( 1 - 9.08T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 + 4.00T + 71T^{2} \) |
| 73 | \( 1 - 1.33T + 73T^{2} \) |
| 79 | \( 1 + 0.797T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 + 0.600T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.52541774354686181086809515139, −7.01384585542203429697544048091, −5.98642408242527740337742354425, −5.36771317067096652503281634507, −4.54088109089268792399450403252, −3.74422930100316423298513111274, −3.39464123886690218244366547006, −2.58787428016204648717408128424, −0.70062826170648893772053426201, 0,
0.70062826170648893772053426201, 2.58787428016204648717408128424, 3.39464123886690218244366547006, 3.74422930100316423298513111274, 4.54088109089268792399450403252, 5.36771317067096652503281634507, 5.98642408242527740337742354425, 7.01384585542203429697544048091, 7.52541774354686181086809515139
