| L(s) = 1 | − 3i·3-s − 3.46i·5-s − 24.2·7-s − 9·9-s + 48i·11-s + 41.5i·13-s − 10.3·15-s + 54·17-s + 4i·19-s + 72.7i·21-s − 173.·23-s + 113·25-s + 27i·27-s + 162. i·29-s − 58.8·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.309i·5-s − 1.30·7-s − 0.333·9-s + 1.31i·11-s + 0.886i·13-s − 0.178·15-s + 0.770·17-s + 0.0482i·19-s + 0.755i·21-s − 1.57·23-s + 0.904·25-s + 0.192i·27-s + 1.04i·29-s − 0.341·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.393621 + 0.512977i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.393621 + 0.512977i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 + 3.46iT - 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 - 48iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 294T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 744. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 252iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 90.0iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 720iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−12.46058063972893470409135604960, −11.72531725301503923798767494187, −10.12230205031406284082650887376, −9.537358612811468881238352832863, −8.297502204857221619160732317865, −7.04212313399442265602401706554, −6.36508021614066942391817636056, −4.86743375806326187067162169522, −3.35834373625043182989910255615, −1.73050350193356663336008338961,
0.27466880455220668469305860575, 2.94003332498992884927593853240, 3.75582166561291065245240375078, 5.57711194019405731154985331277, 6.31825646262571470144875020913, 7.79097429715000858511811352523, 8.900212403660375717467788132052, 9.982375509488921004881448544853, 10.57523233734016695795849167158, 11.74906957341952772114439369007
