| L(s) = 1 | + 1.70i·3-s − 3.02i·5-s + 0.654·7-s + 0.0760·9-s + 1.69i·11-s − 4.95i·13-s + 5.17·15-s + 4.61·17-s − 4.15i·19-s + 1.11i·21-s − 6.69·23-s − 4.15·25-s + 5.25i·27-s − 8.79i·29-s − 6.18·31-s + ⋯ |
| L(s) = 1 | + 0.987i·3-s − 1.35i·5-s + 0.247·7-s + 0.0253·9-s + 0.509i·11-s − 1.37i·13-s + 1.33·15-s + 1.11·17-s − 0.953i·19-s + 0.244i·21-s − 1.39·23-s − 0.830·25-s + 1.01i·27-s − 1.63i·29-s − 1.11·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.553194604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553194604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.70iT - 3T^{2} \) |
| 5 | \( 1 + 3.02iT - 5T^{2} \) |
| 7 | \( 1 - 0.654T + 7T^{2} \) |
| 11 | \( 1 - 1.69iT - 11T^{2} \) |
| 13 | \( 1 + 4.95iT - 13T^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 + 4.15iT - 19T^{2} \) |
| 23 | \( 1 + 6.69T + 23T^{2} \) |
| 29 | \( 1 + 8.79iT - 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 + 6.86iT - 37T^{2} \) |
| 43 | \( 1 - 9.40iT - 43T^{2} \) |
| 47 | \( 1 - 3.05T + 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 5.97iT - 59T^{2} \) |
| 61 | \( 1 + 1.94iT - 61T^{2} \) |
| 67 | \( 1 + 15.3iT - 67T^{2} \) |
| 71 | \( 1 - 1.61T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 2.10T + 79T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 - 7.43T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.901416616938291858623866501428, −7.83742333886208722260425410784, −7.65456459149152252096141060293, −6.06124833725167686888459410169, −5.40414701893894276867368940109, −4.70502940298291084114745441684, −4.15264691698316210640594956153, −3.15830197593960957808324187275, −1.77037388143299745885583641675, −0.51346890550474079116954153274,
1.44348060197827422660427864343, 2.14247889640857147845243812505, 3.32577355188280157734205074302, 3.99113319165689208756976360365, 5.37757673037037541735948580389, 6.20618263780937617195993320653, 6.84350409157748935866044456925, 7.34024777439604810978395570457, 8.059721258928243439645752985648, 8.862985215594735461664481550743
