L(s) = 1 | + 3.50i·7-s − 1.92·11-s − 5.50·13-s − 4.44i·17-s + 7.00i·19-s − 1.10·23-s − 5.47i·29-s + 8.28i·31-s + 0.778·37-s + 2.44i·41-s − 9.55i·43-s + 11.7·47-s − 5.28·49-s + 11.5i·53-s − 9.78·59-s + ⋯ |
L(s) = 1 | + 1.32i·7-s − 0.581·11-s − 1.52·13-s − 1.07i·17-s + 1.60i·19-s − 0.229·23-s − 1.01i·29-s + 1.48i·31-s + 0.127·37-s + 0.381i·41-s − 1.45i·43-s + 1.70·47-s − 0.754·49-s + 1.58i·53-s − 1.27·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3979841532\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3979841532\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.50iT - 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 + 4.44iT - 17T^{2} \) |
| 19 | \( 1 - 7.00iT - 19T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 + 5.47iT - 29T^{2} \) |
| 31 | \( 1 - 8.28iT - 31T^{2} \) |
| 37 | \( 1 - 0.778T + 37T^{2} \) |
| 41 | \( 1 - 2.44iT - 41T^{2} \) |
| 43 | \( 1 + 9.55iT - 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 + 9.78T + 59T^{2} \) |
| 61 | \( 1 + 3.45T + 61T^{2} \) |
| 67 | \( 1 + 5.45iT - 67T^{2} \) |
| 71 | \( 1 + 4.25T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 - 0.386iT - 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
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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
−7.57714605357291094098901560138, −7.29399885683411485683643568872, −6.12610293915014425819676354773, −5.62826904234411289018392627133, −5.00853028852311808834133283733, −4.26792083335364192430068884829, −3.04171077054881427898690024726, −2.55781216983082332613336194314, −1.72064705859625220526321266561, −0.10965882105940201223213821020,
0.890829345714030101716830300944, 2.14619646911895940736399336383, 2.89745257346176776568625706608, 3.90836761181955935151560363147, 4.55598884047717818148275843989, 5.13236202521139929287428054886, 6.09853354065382107639595012754, 6.88448568299344546029456367105, 7.49001319188895656627069048837, 7.82208866448196645758548460317