L(s) = 1 | + (4.24 − 3i)3-s + 5.65·5-s + 16.9i·7-s + (8.99 − 25.4i)9-s + 30i·11-s + 72i·13-s + (24 − 16.9i)15-s + 50.9i·17-s + 25.4·19-s + (50.9 + 71.9i)21-s + 144·23-s − 93·25-s + (−38.1 − 134. i)27-s + 5.65·29-s + 220. i·31-s + ⋯ |
L(s) = 1 | + (0.816 − 0.577i)3-s + 0.505·5-s + 0.916i·7-s + (0.333 − 0.942i)9-s + 0.822i·11-s + 1.53i·13-s + (0.413 − 0.292i)15-s + 0.726i·17-s + 0.307·19-s + (0.529 + 0.748i)21-s + 1.30·23-s − 0.743·25-s + (−0.272 − 0.962i)27-s + 0.0362·29-s + 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.709543348\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.709543348\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.24 + 3i)T \) |
good | 5 | \( 1 - 5.65T + 125T^{2} \) |
| 7 | \( 1 - 16.9iT - 343T^{2} \) |
| 11 | \( 1 - 30iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 72iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 50.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 25.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.65T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 72iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 305. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 576T + 1.03e5T^{2} \) |
| 53 | \( 1 + 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 414iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 504iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 789.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 720T + 3.57e5T^{2} \) |
| 73 | \( 1 + 178T + 3.89e5T^{2} \) |
| 79 | \( 1 - 967. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 438iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 865. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 650T + 9.12e5T^{2} \) |
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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
−11.07844915806142990708049222929, −9.705836982238034401211829012578, −9.171821769045030506534315119467, −8.387982319954147072523445488390, −7.14581181596770401290547447106, −6.46934104582836833493853256325, −5.19581953562491948996588353346, −3.81919092208375403667953494505, −2.39245457328914294874503367372, −1.61474958555642258430873526582,
0.873310423537784602515782137560, 2.69523285749397244256011972089, 3.57526767602722731241816635969, 4.84787677209530728715124635955, 5.86257569031097514380175300652, 7.33650465607363445279330073571, 8.033948267501716579758314343485, 9.086764159154584662266432808009, 9.945123941511747776706424751047, 10.59794948003094254521570782764