| L(s) = 1 | + 4i·3-s + 2i·5-s + 24·7-s + 11·9-s − 44i·11-s + 22i·13-s − 8·15-s + 50·17-s − 44i·19-s + 96i·21-s − 56·23-s + 121·25-s + 152i·27-s + 198i·29-s + 160·31-s + ⋯ |
| L(s) = 1 | + 0.769i·3-s + 0.178i·5-s + 1.29·7-s + 0.407·9-s − 1.20i·11-s + 0.469i·13-s − 0.137·15-s + 0.713·17-s − 0.531i·19-s + 0.997i·21-s − 0.507·23-s + 0.967·25-s + 1.08i·27-s + 1.26i·29-s + 0.926·31-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
| \(L(2)\) |
\(\approx\) |
\(2.227393487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227393487\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4iT - 27T^{2} \) |
| 5 | \( 1 - 2iT - 125T^{2} \) |
| 7 | \( 1 - 24T + 343T^{2} \) |
| 11 | \( 1 + 44iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 22iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 50T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 56T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 160T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 198T + 6.89e4T^{2} \) |
| 43 | \( 1 - 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 528T + 1.03e5T^{2} \) |
| 53 | \( 1 - 242iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 668iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 550iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 188iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 728T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154T + 3.89e5T^{2} \) |
| 79 | \( 1 - 656T + 4.93e5T^{2} \) |
| 83 | \( 1 + 236iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 714T + 7.04e5T^{2} \) |
| 97 | \( 1 + 478T + 9.12e5T^{2} \) |
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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
−11.35047499166022585276366555053, −10.88051738450814344019320489183, −9.866046897648036609719238479419, −8.776236824735504186153362727167, −7.955674051185308519962790024958, −6.68632476866117155317231611359, −5.28513459738486519009620558183, −4.46252931366988852898687069574, −3.16629627615714585627041994211, −1.31508858984382881591029695013,
1.14051792264326923087225674819, 2.20779494334740992285907228242, 4.21794632746671205019696576346, 5.21153801847310990744815398513, 6.57773330060967142659947145430, 7.73862271919934335692220220491, 8.089276694759294786573297308144, 9.626634724317966272834647990296, 10.48425238209486536821393410134, 11.71643329742865489286615263595
