| L(s) = 1 | + (−0.328 + 3.98i)2-s + (−15.7 − 2.61i)4-s + 5.66·5-s − 52.1i·7-s + (15.6 − 62.0i)8-s + (−1.85 + 22.5i)10-s − 106. i·11-s − 122.·13-s + (207. + 17.1i)14-s + (242. + 82.6i)16-s − 122.·17-s + 593. i·19-s + (−89.3 − 14.8i)20-s + (425. + 35.0i)22-s + 546. i·23-s + ⋯ |
| L(s) = 1 | + (−0.0820 + 0.996i)2-s + (−0.986 − 0.163i)4-s + 0.226·5-s − 1.06i·7-s + (0.244 − 0.969i)8-s + (−0.0185 + 0.225i)10-s − 0.881i·11-s − 0.722·13-s + (1.06 + 0.0873i)14-s + (0.946 + 0.322i)16-s − 0.424·17-s + 1.64i·19-s + (−0.223 − 0.0370i)20-s + (0.878 + 0.0723i)22-s + 1.03i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6759784689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6759784689\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.328 - 3.98i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5.66T + 625T^{2} \) |
| 7 | \( 1 + 52.1iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 106. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 122.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 122.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 593. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 546. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 735.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 585. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.28e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.86e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.24e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.75e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.15e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 66.3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.10e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.03e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.70e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.38e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.00e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 3.18e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 4.81e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30866160887086778508915376274, −10.14612736216379084381935739271, −9.616795907349793366794501363615, −8.271462234515167728531387525474, −7.66508761712414799293102608609, −6.57888947694723732245359108084, −5.70657502982566750465668517539, −4.51200718653855275335570624870, −3.45105885980072437710927866408, −1.24632142115537407355125626732,
0.22219667904574917224971608095, 2.06231545625231111476272981836, 2.72873476394137938549147938877, 4.42402239513752708900581067359, 5.19085116729961343819719838765, 6.58335576206291826531812817233, 7.948783019530554569587936820513, 9.007488325073510321654682303725, 9.593943057819786894632105189393, 10.53420863784485197764382547873
