| L(s) = 1 | − 8i·3-s + 5i·5-s − 4·7-s − 37·9-s − 12i·11-s + 58i·13-s + 40·15-s + 66·17-s − 100i·19-s + 32i·21-s + 132·23-s − 25·25-s + 80i·27-s + 90i·29-s − 152·31-s + ⋯ |
| L(s) = 1 | − 1.53i·3-s + 0.447i·5-s − 0.215·7-s − 1.37·9-s − 0.328i·11-s + 1.23i·13-s + 0.688·15-s + 0.941·17-s − 1.20i·19-s + 0.332i·21-s + 1.19·23-s − 0.200·25-s + 0.570i·27-s + 0.576i·29-s − 0.880·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.718057133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718057133\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| good | 3 | \( 1 + 8iT - 27T^{2} \) |
| 7 | \( 1 + 4T + 343T^{2} \) |
| 11 | \( 1 + 12iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 100iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 132T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 152T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 438T + 6.89e4T^{2} \) |
| 43 | \( 1 + 32iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 204T + 1.03e5T^{2} \) |
| 53 | \( 1 - 222iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 420iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 902iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 432T + 3.57e5T^{2} \) |
| 73 | \( 1 + 362T + 3.89e5T^{2} \) |
| 79 | \( 1 - 160T + 4.93e5T^{2} \) |
| 83 | \( 1 - 72iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−8.975287280805214976339080733048, −7.935646030344014569655680072488, −7.20253056205684990320231509499, −6.72193077318100427533334263103, −5.96319200181216474610487650586, −4.86506294395467970151436499211, −3.45617477976136811041237087102, −2.52163419297756270846207188530, −1.52827403572224603466696020579, −0.46513673086794340581250285551,
1.04214109859367355837513261597, 2.79035127196225421676899351848, 3.65300097773270333920247806650, 4.41180159246136435423391767744, 5.43672880766219590606009387896, 5.77795850273166173665972315547, 7.32413559426443359812558459997, 8.142916655522256424869248973162, 9.001991498319209293192737163879, 9.717141575532191684185157017954
