| L(s) = 1 | + (0.256 + 2.22i)5-s − 3.50·7-s + 1.92·11-s − 5.50i·13-s + 4.44·17-s + 7.00i·19-s + 1.10i·23-s + (−4.86 + 1.14i)25-s − 5.47i·29-s + 8.28i·31-s + (−0.900 − 7.78i)35-s − 0.778i·37-s + 2.44i·41-s + 9.55·43-s + 11.7i·47-s + ⋯ |
| L(s) = 1 | + (0.114 + 0.993i)5-s − 1.32·7-s + 0.581·11-s − 1.52i·13-s + 1.07·17-s + 1.60i·19-s + 0.229i·23-s + (−0.973 + 0.228i)25-s − 1.01i·29-s + 1.48i·31-s + (−0.152 − 1.31i)35-s − 0.127i·37-s + 0.381i·41-s + 1.45·43-s + 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9930801184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9930801184\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.256 - 2.22i)T \) |
| good | 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 - 1.92T + 11T^{2} \) |
| 13 | \( 1 + 5.50iT - 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 - 7.00iT - 19T^{2} \) |
| 23 | \( 1 - 1.10iT - 23T^{2} \) |
| 29 | \( 1 + 5.47iT - 29T^{2} \) |
| 31 | \( 1 - 8.28iT - 31T^{2} \) |
| 37 | \( 1 + 0.778iT - 37T^{2} \) |
| 41 | \( 1 - 2.44iT - 41T^{2} \) |
| 43 | \( 1 - 9.55T + 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 9.78T + 59T^{2} \) |
| 61 | \( 1 - 3.45T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 + 4.25T + 71T^{2} \) |
| 73 | \( 1 - 7.27iT - 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 4.25iT - 83T^{2} \) |
| 89 | \( 1 + 0.386iT - 89T^{2} \) |
| 97 | \( 1 - 9.29iT - 97T^{2} \) |
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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
−9.329311373774987170940814648401, −7.987665728478128217065244643638, −7.67900673170262126901960423378, −6.63453490153124954996453104441, −6.03453472721307020355253217146, −5.55153700222195011344911741890, −4.05238399922609455349339191232, −3.23800704132265525515866450873, −2.87261020533850691929509957629, −1.29803643348209330358085973103,
0.33186191198220118403777224063, 1.57421901414967355904167024539, 2.76846867665260107988079743065, 3.81951325651428406277024504402, 4.49423936953548309814188569069, 5.42285933287672922899705818796, 6.30725211329059521804674302037, 6.85305793920337279729172865844, 7.70235897641295140543939220654, 8.809050878371317442581083449615
