| L(s) = 1 | − 1.41i·3-s + (1 − 2i)5-s + 4.82i·7-s + 0.999·9-s + 4.82·11-s − 0.585i·13-s + (−2.82 − 1.41i)15-s + 2.82i·17-s + 19-s + 6.82·21-s + 7.65i·23-s + (−3 − 4i)25-s − 5.65i·27-s − 3.65·29-s + 6.82·31-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + (0.447 − 0.894i)5-s + 1.82i·7-s + 0.333·9-s + 1.45·11-s − 0.162i·13-s + (−0.730 − 0.365i)15-s + 0.685i·17-s + 0.229·19-s + 1.49·21-s + 1.59i·23-s + (−0.600 − 0.800i)25-s − 1.08i·27-s − 0.679·29-s + 1.22·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82023 - 0.429698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82023 - 0.429698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.585iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 23 | \( 1 - 7.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 0.585iT - 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 8.82iT - 43T^{2} \) |
| 47 | \( 1 + 0.828iT - 47T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 9.89iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 1.17iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 1.07iT - 97T^{2} \) |
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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
−9.962646110482734127952223980880, −9.253311181064704564274439393604, −8.677829895631238343481685549504, −7.81573691430590541813983905262, −6.58409211853966635224205939266, −5.91250364726114413921427842003, −5.11442390211032529899716056014, −3.76337759880744972274493669689, −2.15046904564781886536363772481, −1.38487909258119473222844343303,
1.24243521776005975096593263135, 3.05674410845415407159707716459, 4.11317253808898792369126266616, 4.55124669646651139632100676316, 6.22159150786997629898258269138, 6.91371484383468952760536954479, 7.54459091350667474916127375640, 8.990931909751911179727513175603, 9.836387558094607040030123917911, 10.27622517217309956868576988164
