| L(s) = 1 | + 3.38i·2-s − 3.43·4-s + 11.1i·5-s + 15.4i·8-s − 37.8·10-s − 79.6·16-s + 87.3i·17-s − 102.·19-s − 38.4i·20-s − 121. i·23-s − 125.·25-s + 337.·31-s − 146. i·32-s − 295.·34-s − 345. i·38-s + ⋯ |
| L(s) = 1 | + 1.19i·2-s − 0.429·4-s + 0.999i·5-s + 0.681i·8-s − 1.19·10-s − 1.24·16-s + 1.24i·17-s − 1.23·19-s − 0.429i·20-s − 1.10i·23-s − 1.00·25-s + 1.95·31-s − 0.806i·32-s − 1.48·34-s − 1.47i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(-1.44977i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-1.44977i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 11.1iT \) |
| good | 2 | \( 1 - 3.38iT - 8T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 87.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 121. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 337.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 - 545. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 706. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 943.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−13.64780204490997389981149446241, −12.33093622816217916630777960765, −11.03684341006034730693905501761, −10.28364660916053189175788821847, −8.657837297765065797154187145747, −7.79561661449116956028900453729, −6.58637689037998824063191942587, −6.06058496397866354045731673376, −4.35202171592028220192543487350, −2.48041639092386491450575667598,
0.72383537371945490398221170596, 2.25995373726390120186956476055, 3.86645918455041812306170305149, 5.11089550279912590835415889996, 6.76715385890649328285736993788, 8.291996963272231124834197752744, 9.396253667585529324086830624659, 10.20991386503770028181743532434, 11.48730027676362034014371624909, 12.04295476909782200434221703930
