| L(s) = 1 | + (−193. + 193. i)3-s + (1.03e4 − 1.03e4i)5-s − 1.08e5·7-s + 4.56e5i·9-s + (1.04e6 + 1.04e6i)11-s + (1.27e6 + 1.27e6i)13-s + 4.00e6i·15-s + 1.95e7·17-s + (2.72e7 − 2.72e7i)19-s + (2.09e7 − 2.09e7i)21-s − 1.76e8·23-s + 2.96e7i·25-s + (−1.91e8 − 1.91e8i)27-s + (3.27e8 + 3.27e8i)29-s − 1.40e9i·31-s + ⋯ |
| L(s) = 1 | + (−0.265 + 0.265i)3-s + (0.662 − 0.662i)5-s − 0.921·7-s + 0.859i·9-s + (0.590 + 0.590i)11-s + (0.264 + 0.264i)13-s + 0.351i·15-s + 0.808·17-s + (0.579 − 0.579i)19-s + (0.244 − 0.244i)21-s − 1.19·23-s + 0.121i·25-s + (−0.493 − 0.493i)27-s + (0.550 + 0.550i)29-s − 1.58i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.351939787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351939787\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (193. - 193. i)T - 5.31e5iT^{2} \) |
| 5 | \( 1 + (-1.03e4 + 1.03e4i)T - 2.44e8iT^{2} \) |
| 7 | \( 1 + 1.08e5T + 1.38e10T^{2} \) |
| 11 | \( 1 + (-1.04e6 - 1.04e6i)T + 3.13e12iT^{2} \) |
| 13 | \( 1 + (-1.27e6 - 1.27e6i)T + 2.32e13iT^{2} \) |
| 17 | \( 1 - 1.95e7T + 5.82e14T^{2} \) |
| 19 | \( 1 + (-2.72e7 + 2.72e7i)T - 2.21e15iT^{2} \) |
| 23 | \( 1 + 1.76e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-3.27e8 - 3.27e8i)T + 3.53e17iT^{2} \) |
| 31 | \( 1 + 1.40e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + (3.08e9 - 3.08e9i)T - 6.58e18iT^{2} \) |
| 41 | \( 1 + 1.10e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + (-4.13e9 - 4.13e9i)T + 3.99e19iT^{2} \) |
| 47 | \( 1 + 7.39e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + (-2.73e10 + 2.73e10i)T - 4.91e20iT^{2} \) |
| 59 | \( 1 + (4.52e9 + 4.52e9i)T + 1.77e21iT^{2} \) |
| 61 | \( 1 + (2.45e9 + 2.45e9i)T + 2.65e21iT^{2} \) |
| 67 | \( 1 + (-2.12e9 + 2.12e9i)T - 8.18e21iT^{2} \) |
| 71 | \( 1 - 5.19e10T + 1.64e22T^{2} \) |
| 73 | \( 1 - 2.15e11iT - 2.29e22T^{2} \) |
| 79 | \( 1 - 3.83e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (3.95e11 - 3.95e11i)T - 1.06e23iT^{2} \) |
| 89 | \( 1 - 7.22e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 1.01e12T + 6.93e23T^{2} \) |
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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
−11.41839418335350645588931716816, −9.999432562827996077839573207236, −9.632103508820633574211989792180, −8.371856491171131391214034266990, −7.03407599228208013822442991181, −5.87810343940263912792803452124, −4.97676153249777867605610088026, −3.75846682622996893194035718883, −2.26657585130314583749171827588, −1.12221101310989833881708057402,
0.31109591323539581068374754805, 1.44316948718467190781367174571, 2.97488740907713103196483070921, 3.75962685331996870989978881268, 5.79516889089440351854225916556, 6.23443868316401921385969617136, 7.27180552563325844921121414187, 8.775441686303417904591261472247, 9.816359112293871563222766098507, 10.52656738200804197946485776646
