| L(s) = 1 | − 2i·3-s + 6i·5-s − 20·7-s + 23·9-s + 14i·11-s − 54i·13-s + 12·15-s − 66·17-s − 162i·19-s + 40i·21-s − 172·23-s + 89·25-s − 100i·27-s + 2i·29-s − 128·31-s + ⋯ |
| L(s) = 1 | − 0.384i·3-s + 0.536i·5-s − 1.07·7-s + 0.851·9-s + 0.383i·11-s − 1.15i·13-s + 0.206·15-s − 0.941·17-s − 1.95i·19-s + 0.415i·21-s − 1.55·23-s + 0.711·25-s − 0.712i·27-s + 0.0128i·29-s − 0.741·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7737000573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7737000573\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 27T^{2} \) |
| 5 | \( 1 - 6iT - 125T^{2} \) |
| 7 | \( 1 + 20T + 343T^{2} \) |
| 11 | \( 1 - 14iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 54iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 162iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 172T + 1.21e4T^{2} \) |
| 29 | \( 1 - 2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 128T + 2.97e4T^{2} \) |
| 37 | \( 1 - 158iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 202T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 408T + 1.03e5T^{2} \) |
| 53 | \( 1 + 690iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 322iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 298iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 202iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 700T + 3.57e5T^{2} \) |
| 73 | \( 1 - 418T + 3.89e5T^{2} \) |
| 79 | \( 1 - 744T + 4.93e5T^{2} \) |
| 83 | \( 1 - 678iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 82T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−11.13766477335058199241036197796, −10.20933697575901816904992928190, −9.478500063167690374169938011415, −8.180534893498519379782528293148, −6.92050518766229709018673813712, −6.57501172101635111372119675474, −4.98543775143759948283876500824, −3.53848057744511487794138093367, −2.29322460015844818711690119812, −0.29291374160898331249122885786,
1.72540361878266596083152973375, 3.62195825889249314631442671288, 4.45486216720036983971795561980, 5.94126336802432237634777685809, 6.82852509645970742603065821296, 8.143870539090073500484143449758, 9.275373929671376793805333079964, 9.854070737280347293667452203539, 10.84838661678314578719945356647, 12.12532635346900776171116018931
