| L(s) = 1 | + 0.329i·2-s + 1.89·4-s + (−2.08 − 0.805i)5-s + 3.70i·7-s + 1.28i·8-s + (0.265 − 0.687i)10-s + 3.31·11-s − i·13-s − 1.21·14-s + 3.36·16-s + 4.36i·17-s − 5.21·19-s + (−3.94 − 1.52i)20-s + 1.09i·22-s + 4.92i·23-s + ⋯ |
| L(s) = 1 | + 0.232i·2-s + 0.945·4-s + (−0.932 − 0.360i)5-s + 1.39i·7-s + 0.453i·8-s + (0.0839 − 0.217i)10-s + 0.998·11-s − 0.277i·13-s − 0.325·14-s + 0.840·16-s + 1.05i·17-s − 1.19·19-s + (−0.882 − 0.340i)20-s + 0.232i·22-s + 1.02i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29114 + 0.885438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29114 + 0.885438i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.08 + 0.805i)T \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - 0.329iT - 2T^{2} \) |
| 7 | \( 1 - 3.70iT - 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 17 | \( 1 - 4.36iT - 17T^{2} \) |
| 19 | \( 1 + 5.21T + 19T^{2} \) |
| 23 | \( 1 - 4.92iT - 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 - 0.0981T + 31T^{2} \) |
| 37 | \( 1 + 2.92iT - 37T^{2} \) |
| 41 | \( 1 - 0.749T + 41T^{2} \) |
| 43 | \( 1 - 3.78iT - 43T^{2} \) |
| 47 | \( 1 + 5.67iT - 47T^{2} \) |
| 53 | \( 1 + 2.19iT - 53T^{2} \) |
| 59 | \( 1 - 0.108T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 9.56iT - 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 16.7iT - 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + 4.10iT - 97T^{2} \) |
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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
−11.12090421585154049601103713289, −10.05252902806239979985962130549, −8.666996064462306766222899826683, −8.451079019341726138041554563013, −7.26852646774455283629802263713, −6.31593230122065880860675323293, −5.57953905609053955568677521438, −4.23239652696001953092693618639, −3.03817710069561507748669323118, −1.72072258959728659975176670436,
0.935998098383522678037904896440, 2.66697146143296278338786802829, 3.83283386693435078363065734459, 4.53235185296908404528677301557, 6.52389838581936288229935878461, 6.81677612441191762873875629102, 7.68079284694963557769389035238, 8.662373936980562330924346909152, 10.02440868606663447467838949940, 10.65203342114629512508118292466
