| L(s) = 1 | − 8.49·3-s + 59.7i·5-s − 483. i·7-s − 656.·9-s − 1.41e3·11-s − 3.45e3i·13-s − 507. i·15-s − 3.05e3·17-s − 968.·19-s + 4.10e3i·21-s − 3.31e3i·23-s + 1.20e4·25-s + 1.17e4·27-s + 2.63e4i·29-s + 2.71e4i·31-s + ⋯ |
| L(s) = 1 | − 0.314·3-s + 0.477i·5-s − 1.40i·7-s − 0.901·9-s − 1.06·11-s − 1.57i·13-s − 0.150i·15-s − 0.622·17-s − 0.141·19-s + 0.443i·21-s − 0.272i·23-s + 0.771·25-s + 0.598·27-s + 1.08i·29-s + 0.909i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.246424 - 0.590006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246424 - 0.590006i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 8.49T + 729T^{2} \) |
| 5 | \( 1 - 59.7iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 483. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.41e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.45e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 3.05e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 968.T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.31e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.63e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.71e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.60e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 6.86e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 9.28e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.59e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 8.66e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.28e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 1.89e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 3.19e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.96e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.39e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.64e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 8.02e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 5.41e4T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.10e6T + 8.32e11T^{2} \) |
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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
−15.03040559907613783924404657839, −13.82401923957213774144243897322, −12.72306071203828017018182883567, −10.86299077064413608191794950223, −10.44931328287534250275443565892, −8.276513698884621995220197163758, −6.93769845962235968085731887073, −5.20964117883523496428053849703, −3.12993242360962344080984799079, −0.33315260259521740639873325378,
2.39668160667571530132510763994, 4.93309883891185644006643832140, 6.22123765811061321258980086081, 8.337706222879000608607081849135, 9.345022556518291158593230223827, 11.24946814759943656727139287251, 12.11768400546339086655172365629, 13.44327702396283101856897251681, 14.90634986230978146986573876904, 16.00000409175268542553821218825
