| L(s) = 1 | + 3.57·3-s − 2.23i·5-s + 10.2i·7-s + 3.78·9-s − 7.58i·11-s − 5.25·13-s − 7.99i·15-s + 28.1i·17-s + 8.29i·19-s + 36.5i·21-s + (20.8 − 9.69i)23-s − 5.00·25-s − 18.6·27-s − 41.1·29-s − 27.0·31-s + ⋯ |
| L(s) = 1 | + 1.19·3-s − 0.447i·5-s + 1.46i·7-s + 0.420·9-s − 0.689i·11-s − 0.404·13-s − 0.533i·15-s + 1.65i·17-s + 0.436i·19-s + 1.74i·21-s + (0.906 − 0.421i)23-s − 0.200·25-s − 0.690·27-s − 1.41·29-s − 0.873·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.985263933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.985263933\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-20.8 + 9.69i)T \) |
| good | 3 | \( 1 - 3.57T + 9T^{2} \) |
| 7 | \( 1 - 10.2iT - 49T^{2} \) |
| 11 | \( 1 + 7.58iT - 121T^{2} \) |
| 13 | \( 1 + 5.25T + 169T^{2} \) |
| 17 | \( 1 - 28.1iT - 289T^{2} \) |
| 19 | \( 1 - 8.29iT - 361T^{2} \) |
| 29 | \( 1 + 41.1T + 841T^{2} \) |
| 31 | \( 1 + 27.0T + 961T^{2} \) |
| 37 | \( 1 - 62.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 23.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 27.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6.43T + 2.20e3T^{2} \) |
| 53 | \( 1 - 4.28iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 67.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 83.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 7.91iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 71.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 76.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 84.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 45.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 170. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 158. iT - 9.40e3T^{2} \) |
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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
−9.084316152670798552150408655785, −8.474556719297214318428355142640, −8.225840886308098891241206520712, −7.10659950879269571302479028757, −5.85534241707411123297860167793, −5.50889282164276554855291260571, −4.17362566719051677886027562451, −3.28229105686409978031391851148, −2.45193631869849255763793599728, −1.57271643080993494055354333695,
0.39691555354390841236068968955, 1.89766202310115606380403235738, 2.86492417022157264886995808854, 3.64548088710172152780742071824, 4.47058679610907011884828929228, 5.46683195399025027289553691191, 6.92110570004994120530694700086, 7.43107704067360490000120377277, 7.65873415024282918513528582070, 9.122486177542584517451485891685
