| L(s) = 1 | − 1.92i·3-s − 2.16·5-s + (−1.74 + 1.99i)7-s − 0.716·9-s + 5.04·11-s − 13-s + 4.17i·15-s + 0.687i·17-s + 1.09i·19-s + (3.84 + 3.35i)21-s + 3.80i·23-s − 0.304·25-s − 4.40i·27-s − 6.62i·29-s − 8.65·31-s + ⋯ |
| L(s) = 1 | − 1.11i·3-s − 0.969·5-s + (−0.657 + 0.753i)7-s − 0.238·9-s + 1.52·11-s − 0.277·13-s + 1.07i·15-s + 0.166i·17-s + 0.251i·19-s + (0.838 + 0.732i)21-s + 0.792i·23-s − 0.0609·25-s − 0.847i·27-s − 1.23i·29-s − 1.55·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5088132506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5088132506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.74 - 1.99i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.92iT - 3T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 17 | \( 1 - 0.687iT - 17T^{2} \) |
| 19 | \( 1 - 1.09iT - 19T^{2} \) |
| 23 | \( 1 - 3.80iT - 23T^{2} \) |
| 29 | \( 1 + 6.62iT - 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 + 9.25iT - 37T^{2} \) |
| 41 | \( 1 + 6.30iT - 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 + 2.89iT - 53T^{2} \) |
| 59 | \( 1 - 11.6iT - 59T^{2} \) |
| 61 | \( 1 + 1.96T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 5.07iT - 71T^{2} \) |
| 73 | \( 1 - 2.08iT - 73T^{2} \) |
| 79 | \( 1 - 0.772iT - 79T^{2} \) |
| 83 | \( 1 - 5.41iT - 83T^{2} \) |
| 89 | \( 1 - 3.65iT - 89T^{2} \) |
| 97 | \( 1 - 2.48iT - 97T^{2} \) |
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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
−8.257118189851736979005609329459, −7.41181730520105228655029729370, −7.10056563363711176660044463054, −6.14535202043904767585459851847, −5.62866617369737163893257597644, −4.11033204860625410681345523068, −3.71193196580043399539424752717, −2.42749355671504859813166391335, −1.50082490306589544237913273192, −0.17393860761982839905140765498,
1.31126299495054560895354452011, 3.12040316081107264575255129214, 3.71942964673680034680407290766, 4.30954408210685666686310319282, 4.93238426445441147058509186416, 6.21351399397616541175852202491, 6.95046002670019381840393943064, 7.52112097341323721027372451283, 8.596426631760563868218691342279, 9.264886453303685561754033963352
