| L(s) = 1 | − 1.23i·3-s + 2.23·5-s + 5.23i·7-s + 1.47·9-s − 2.76i·15-s + 6.47·21-s − 7.70i·23-s + 5.00·25-s − 5.52i·27-s − 6·29-s + 11.7i·35-s + 4.47·41-s + 6.76i·43-s + 3.29·45-s + 0.291i·47-s + ⋯ |
| L(s) = 1 | − 0.713i·3-s + 0.999·5-s + 1.97i·7-s + 0.490·9-s − 0.713i·15-s + 1.41·21-s − 1.60i·23-s + 1.00·25-s − 1.06i·27-s − 1.11·29-s + 1.97i·35-s + 0.698·41-s + 1.03i·43-s + 0.490·45-s + 0.0425i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 3 | \( 1 + 1.23iT - 3T^{2} \) |
| 7 | \( 1 - 5.23iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 7.70iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 6.76iT - 43T^{2} \) |
| 47 | \( 1 - 0.291iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4.29iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.91305181615955581374395895661, −10.71476697737743928414110810542, −9.547742452278862363388597395928, −8.934572892443817482566157036057, −7.88735130799031441078529146259, −6.50484342918042029504547676519, −5.93781900484777146602802890623, −4.83253584996688005409165278981, −2.70872896647222072483784235026, −1.82479599878298816971131228526,
1.44690429251901538790744228020, 3.52816577984308970527095517834, 4.41863975025615865808637163119, 5.57001803235916423991812242157, 6.93029773178061746386814311319, 7.59929071574645992488737823723, 9.210898584225404758638746022017, 9.905508536899981604685191333924, 10.51325240086142331296691235590, 11.25063408779311638466997244276
