| L(s) = 1 | − 0.552·2-s − 2.06·3-s − 1.69·4-s + 3.05·5-s + 1.14·6-s − 3.72·7-s + 2.04·8-s + 1.27·9-s − 1.68·10-s − 2.20·11-s + 3.50·12-s + 2.13·13-s + 2.05·14-s − 6.31·15-s + 2.26·16-s − 1.44·17-s − 0.704·18-s − 3.68·19-s − 5.17·20-s + 7.69·21-s + 1.21·22-s − 23-s − 4.22·24-s + 4.32·25-s − 1.17·26-s + 3.56·27-s + 6.31·28-s + ⋯ |
| L(s) = 1 | − 0.390·2-s − 1.19·3-s − 0.847·4-s + 1.36·5-s + 0.466·6-s − 1.40·7-s + 0.721·8-s + 0.425·9-s − 0.533·10-s − 0.665·11-s + 1.01·12-s + 0.592·13-s + 0.549·14-s − 1.63·15-s + 0.565·16-s − 0.351·17-s − 0.166·18-s − 0.846·19-s − 1.15·20-s + 1.68·21-s + 0.259·22-s − 0.208·23-s − 0.861·24-s + 0.865·25-s − 0.231·26-s + 0.686·27-s + 1.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5807692096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5807692096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.552T + 2T^{2} \) |
| 3 | \( 1 + 2.06T + 3T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 - 2.13T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 - 8.63T + 37T^{2} \) |
| 41 | \( 1 - 9.34T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 + 0.542T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 + 4.94T + 97T^{2} \) |
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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
−10.47326568420741086640342450898, −9.648518134754606847819855034341, −9.173725995437046185218944823353, −8.018343836538494590664309277911, −6.51414648269798103022235787781, −6.09041539988016934392213045149, −5.31324132658385777053343654362, −4.19428557177685333967995247575, −2.58099701622677513943113391642, −0.72228352570196251496121409999,
0.72228352570196251496121409999, 2.58099701622677513943113391642, 4.19428557177685333967995247575, 5.31324132658385777053343654362, 6.09041539988016934392213045149, 6.51414648269798103022235787781, 8.018343836538494590664309277911, 9.173725995437046185218944823353, 9.648518134754606847819855034341, 10.47326568420741086640342450898
