L(s) = 1 | + (1.36 − 1.46i)2-s − 1.73·3-s + (−0.267 − 3.99i)4-s + 7.98i·5-s + (−2.36 + 2.53i)6-s − 2.13i·7-s + (−6.19 − 5.06i)8-s + 2.99·9-s + (11.6 + 10.9i)10-s − 8·11-s + (0.464 + 6.91i)12-s − 11.6i·13-s + (−3.12 − 2.92i)14-s − 13.8i·15-s + (−15.8 + 2.13i)16-s + 11.8·17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.730i)2-s − 0.577·3-s + (−0.0669 − 0.997i)4-s + 1.59i·5-s + (−0.394 + 0.421i)6-s − 0.305i·7-s + (−0.774 − 0.632i)8-s + 0.333·9-s + (1.16 + 1.09i)10-s − 0.727·11-s + (0.0386 + 0.576i)12-s − 0.898i·13-s + (−0.223 − 0.208i)14-s − 0.921i·15-s + (−0.991 + 0.133i)16-s + 0.697·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.985254 - 0.351206i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.985254 - 0.351206i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 1.46i)T \) |
| 3 | \( 1 + 1.73T \) |
good | 5 | \( 1 - 7.98iT - 25T^{2} \) |
| 7 | \( 1 + 2.13iT - 49T^{2} \) |
| 11 | \( 1 + 8T + 121T^{2} \) |
| 13 | \( 1 + 11.6iT - 169T^{2} \) |
| 17 | \( 1 - 11.8T + 289T^{2} \) |
| 19 | \( 1 - 14.9T + 361T^{2} \) |
| 23 | \( 1 - 4.27iT - 529T^{2} \) |
| 29 | \( 1 + 0.573iT - 841T^{2} \) |
| 31 | \( 1 - 57.4iT - 961T^{2} \) |
| 37 | \( 1 + 27.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 31.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 28.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 59.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 31.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 52.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 59.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 84.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 42.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.42T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 67.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 97.1T + 9.40e3T^{2} \) |
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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
−17.87925667205782554548000445744, −15.80553222971354371385550171985, −14.70441004541540263996554334350, −13.57886089834994631130068568919, −12.09426334571972855301389625462, −10.76293102666116434721923976255, −10.19749434703203209978610563842, −7.16093454727449572360219821350, −5.55622303442588941650820156498, −3.19126847826640312997642897644,
4.61158122069529405690835623875, 5.78138868873153761470703667285, 7.82621438104503750076123410916, 9.284699578591150270908648951262, 11.74012896545958677224446927740, 12.63206147962328451585534254910, 13.71617300939241384390984271921, 15.48206265842981341922040834870, 16.44775431237460235918000820954, 17.09365488182585908799684007631