| L(s) = 1 | − 2-s + 4-s + (2.21 − 0.288i)5-s + 3.14i·7-s − 8-s + (−2.21 + 0.288i)10-s + 0.908·11-s − 2.22·13-s − 3.14i·14-s + 16-s + 2.10·17-s + 4.16i·19-s + (2.21 − 0.288i)20-s − 0.908·22-s − 7.66·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.991 − 0.129i)5-s + 1.19i·7-s − 0.353·8-s + (−0.701 + 0.0912i)10-s + 0.274·11-s − 0.616·13-s − 0.841i·14-s + 0.250·16-s + 0.509·17-s + 0.955i·19-s + (0.495 − 0.0645i)20-s − 0.193·22-s − 1.59·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.215155241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215155241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.21 + 0.288i)T \) |
| 37 | \( 1 + (-1.10 - 5.98i)T \) |
| good | 7 | \( 1 - 3.14iT - 7T^{2} \) |
| 11 | \( 1 - 0.908T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 - 4.16iT - 19T^{2} \) |
| 23 | \( 1 + 7.66T + 23T^{2} \) |
| 29 | \( 1 - 2.69iT - 29T^{2} \) |
| 31 | \( 1 + 5.96iT - 31T^{2} \) |
| 41 | \( 1 + 2.32T + 41T^{2} \) |
| 43 | \( 1 - 5.72T + 43T^{2} \) |
| 47 | \( 1 - 8.89iT - 47T^{2} \) |
| 53 | \( 1 - 9.37iT - 53T^{2} \) |
| 59 | \( 1 + 5.55iT - 59T^{2} \) |
| 61 | \( 1 + 3.16iT - 61T^{2} \) |
| 67 | \( 1 - 7.64iT - 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 2.14iT - 73T^{2} \) |
| 79 | \( 1 - 3.35iT - 79T^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 + 8.35iT - 89T^{2} \) |
| 97 | \( 1 - 5.14T + 97T^{2} \) |
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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
−8.958632091763866320655505076573, −8.187857041311468309819848563638, −7.58292010496930204820352875805, −6.44213046292862068692446823275, −5.91644524805766975795888986276, −5.38290393875308155409637894919, −4.22912974607480966585309038872, −2.92127126972371936208676568488, −2.20489699628674674146932767275, −1.36248126829641796116360337046,
0.45626486441023958219601270245, 1.61088628408445634365650244488, 2.49929399357416465622982780121, 3.58499942934263281348922247990, 4.55042617929509299587048802968, 5.52422361965004973172780805600, 6.31007172995617912934281972807, 7.08913238619905719431743291021, 7.51719558263202985875801984613, 8.502041326597532616141914448352
