| L(s) = 1 | + 3.48e3i·3-s − 3.49e4i·5-s + (1.89e5 − 8.01e5i)7-s − 7.39e6·9-s − 1.79e7·11-s − 1.12e7i·13-s + 1.21e8·15-s − 4.11e8i·17-s − 4.61e8i·19-s + (2.79e9 + 6.61e8i)21-s + 3.11e9·23-s − 1.22e9·25-s − 9.10e9i·27-s + 1.41e10·29-s − 1.84e9i·31-s + ⋯ |
| L(s) = 1 | + 1.59i·3-s − 0.447i·5-s + (0.230 − 0.973i)7-s − 1.54·9-s − 0.922·11-s − 0.178i·13-s + 0.713·15-s − 1.00i·17-s − 0.516i·19-s + (1.55 + 0.367i)21-s + 0.913·23-s − 0.199·25-s − 0.869i·27-s + 0.822·29-s − 0.0671i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.6681065924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6681065924\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.49e4iT \) |
| 7 | \( 1 + (-1.89e5 + 8.01e5i)T \) |
| good | 3 | \( 1 - 3.48e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 + 1.79e7T + 3.79e14T^{2} \) |
| 13 | \( 1 + 1.12e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + 4.11e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 4.61e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 - 3.11e9T + 1.15e19T^{2} \) |
| 29 | \( 1 - 1.41e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 1.84e9iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 8.59e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 1.00e10iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 4.77e11T + 7.38e22T^{2} \) |
| 47 | \( 1 - 2.27e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 9.87e11T + 1.37e24T^{2} \) |
| 59 | \( 1 + 1.56e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 5.85e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 1.34e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 8.97e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + 9.51e11iT - 1.22e26T^{2} \) |
| 79 | \( 1 - 4.54e11T + 3.68e26T^{2} \) |
| 83 | \( 1 - 3.93e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 1.38e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 4.48e13iT - 6.52e27T^{2} \) |
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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
−10.73489760717443211968049244577, −10.12953448196122362384300826324, −9.202849178905309673776923011164, −8.196833094644836344552169643080, −6.96271353357746638125214594334, −5.19549076114114725543669963470, −4.83179421631736433550090576511, −3.73679673867956100392704766479, −2.72475962265147888648657206590, −0.913700808655373718404111864269,
0.14123267779700862085990566254, 1.50285207852560404824443653985, 2.22830282897916677390156898905, 3.21170368451305089478376621239, 5.13814391486969067286941551910, 6.13090009282351375029235651698, 6.95826835151767059592162530885, 8.025113229118671316123974822481, 8.639021034247148687668120236401, 10.22360136453236501051053418598
