| L(s) = 1 | − 2i·3-s + (5 + 10i)5-s − 26i·7-s + 23·9-s − 28·11-s + 12i·13-s + (20 − 10i)15-s − 64i·17-s + 60·19-s − 52·21-s − 58i·23-s + (−75 + 100i)25-s − 100i·27-s + 90·29-s + 128·31-s + ⋯ |
| L(s) = 1 | − 0.384i·3-s + (0.447 + 0.894i)5-s − 1.40i·7-s + 0.851·9-s − 0.767·11-s + 0.256i·13-s + (0.344 − 0.172i)15-s − 0.913i·17-s + 0.724·19-s − 0.540·21-s − 0.525i·23-s + (−0.599 + 0.800i)25-s − 0.712i·27-s + 0.576·29-s + 0.741·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.965053074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965053074\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-5 - 10i)T \) |
| good | 3 | \( 1 + 2iT - 27T^{2} \) |
| 7 | \( 1 + 26iT - 343T^{2} \) |
| 11 | \( 1 + 28T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 64iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 60T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128T + 2.97e4T^{2} \) |
| 37 | \( 1 + 236iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 242T + 6.89e4T^{2} \) |
| 43 | \( 1 + 362iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 226iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 108iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 20T + 2.05e5T^{2} \) |
| 61 | \( 1 + 542T + 2.26e5T^{2} \) |
| 67 | \( 1 + 434iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 632iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 720T + 4.93e5T^{2} \) |
| 83 | \( 1 - 478iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 490T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3iT - 9.12e5T^{2} \) |
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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
−10.75934414787866686844217665164, −10.30126145728660164995251069892, −9.421919743358524568654159604041, −7.74916666265738274204267833242, −7.22267051190744680595668163194, −6.43977473333759702749282207859, −4.95820813759509965497682588817, −3.72149822382090576417837097875, −2.35273775983510710289686278097, −0.78098985513471032648347628833,
1.41694947023582588877187527756, 2.81833708498260763512617423949, 4.47427146756510806226847639870, 5.35296152514369661033891437439, 6.18631286059855320480283079748, 7.80332822410156166923342569114, 8.628415232871142687758894902185, 9.561829084588273935060140465164, 10.17040937531482725483479709157, 11.43841925571407403799072145603
