L(s) = 1 | + 3i·3-s + 5.86·5-s + (18.3 + 2.64i)7-s − 9·9-s − 34.5·11-s + 55.8·13-s + 17.5i·15-s + 2.77i·17-s − 67.8i·19-s + (−7.94 + 54.9i)21-s − 176. i·23-s − 90.6·25-s − 27i·27-s − 116. i·29-s + 312.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.524·5-s + (0.989 + 0.143i)7-s − 0.333·9-s − 0.946·11-s + 1.19·13-s + 0.302i·15-s + 0.0395i·17-s − 0.819i·19-s + (−0.0826 + 0.571i)21-s − 1.59i·23-s − 0.724·25-s − 0.192i·27-s − 0.743i·29-s + 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.518363525\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.518363525\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (-18.3 - 2.64i)T \) |
good | 5 | \( 1 - 5.86T + 125T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.77iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 67.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 176. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 116. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 280. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 15.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 6.34T + 1.03e5T^{2} \) |
| 53 | \( 1 - 23.2iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 288. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 475. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 473. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 796. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 877. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 33.8iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 700. iT - 9.12e5T^{2} \) |
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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
−9.106534507785525608134684906705, −8.409300572712208850209026484794, −7.85003936573859149093646859610, −6.53049359149655636861666186500, −5.80220308296943781193908359653, −4.88929111208320296923226888666, −4.25459164680258638660593058663, −2.91238609440279625774510119709, −2.02062177891702435710470051133, −0.62198814809452752348625455023,
1.11864283460175819728115921365, 1.82456516987459920474647470601, 3.01063862823282269651508255668, 4.17406854034897631331054464519, 5.35513996622629108371618997304, 5.84865091735134172309418283510, 6.85891923926160897093153602782, 8.018970953557332640166702160102, 8.072614124066005040486237396850, 9.252183531913923089902263045125