L(s) = 1 | − 2.62i·3-s + 11.5i·5-s − 23.0i·7-s + 20.1·9-s − 1.19i·11-s + 58.3·13-s + 30.2·15-s + (−54.8 − 43.6i)17-s + 138.·19-s − 60.4·21-s − 180. i·23-s − 7.98·25-s − 123. i·27-s − 115. i·29-s + 210. i·31-s + ⋯ |
L(s) = 1 | − 0.504i·3-s + 1.03i·5-s − 1.24i·7-s + 0.745·9-s − 0.0327i·11-s + 1.24·13-s + 0.520·15-s + (−0.782 − 0.622i)17-s + 1.67·19-s − 0.627·21-s − 1.64i·23-s − 0.0638·25-s − 0.880i·27-s − 0.736i·29-s + 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.68076 - 0.586758i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68076 - 0.586758i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (54.8 + 43.6i)T \) |
good | 3 | \( 1 + 2.62iT - 27T^{2} \) |
| 5 | \( 1 - 11.5iT - 125T^{2} \) |
| 7 | \( 1 + 23.0iT - 343T^{2} \) |
| 11 | \( 1 + 1.19iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 58.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 115. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 210. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 210. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 297. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 706.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 489. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 167.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 818. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 110. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 118.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 400.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 924. iT - 9.12e5T^{2} \) |
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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
−12.82988161074864512167161257058, −11.42114461196812305156892536426, −10.66869468150050434325391888166, −9.754773591310759185388130246400, −8.121738153984586735638281539542, −7.02608184319071590435899080800, −6.50475756465951423525505010232, −4.47876022020932137234349562861, −3.09807130355150348641933600340, −1.10222334703488268174480279219,
1.52255253395124227206334714495, 3.63545946294933430942593375005, 5.00895300638379963717581892643, 5.94148685814517768947610144992, 7.68294401288215629448384176458, 9.030054106137236379466249608401, 9.339955632610303486266926163705, 10.86431362976493171304382198687, 11.90412383666398273629872678022, 12.83809028253263988130793244945