| L(s) = 1 | − i·2-s − 4-s + 0.561i·5-s − i·7-s + i·8-s + 0.561·10-s − 1.43i·11-s + (−0.561 + 3.56i)13-s − 14-s + 16-s − 5.68·17-s + 2.56i·19-s − 0.561i·20-s − 1.43·22-s − 5.68·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.251i·5-s − 0.377i·7-s + 0.353i·8-s + 0.177·10-s − 0.433i·11-s + (−0.155 + 0.987i)13-s − 0.267·14-s + 0.250·16-s − 1.37·17-s + 0.587i·19-s − 0.125i·20-s − 0.306·22-s − 1.18·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6672855954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6672855954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (0.561 - 3.56i)T \) |
| good | 5 | \( 1 - 0.561iT - 5T^{2} \) |
| 11 | \( 1 + 1.43iT - 11T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 - 2.56iT - 19T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 + 1.68iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.24iT - 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 12.2iT - 59T^{2} \) |
| 61 | \( 1 - 2.56T + 61T^{2} \) |
| 67 | \( 1 - 7.12iT - 67T^{2} \) |
| 71 | \( 1 - 15.3iT - 71T^{2} \) |
| 73 | \( 1 - 7.43iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.684654322148748228030493853632, −8.759084212614117702387245959603, −8.285669142384718736773497877302, −6.99040742126490024625468282418, −6.51806895831005079161044967438, −5.28230985526186378531729862953, −4.37609245092264660481832211416, −3.61608343461578230176173339762, −2.51316754184414403506535082180, −1.46392057190331767220953831121,
0.25206554934669423520573263660, 2.03918087890916176711063274807, 3.22225045847369398849373360800, 4.53911233509606547738407397057, 4.98307165946397463022839154433, 6.15886325572419055347140776746, 6.61310741798320649818039358926, 7.81021867252914182273784907333, 8.189539338820481759939802671797, 9.190292413249923740587369720609
