L(s) = 1 | + (1.35 − 0.785i)2-s + (3.43 + 3.89i)3-s + (−2.76 + 4.79i)4-s + (7.73 + 2.60i)6-s + (29.6 − 17.1i)7-s + 21.2i·8-s + (−3.40 + 26.7i)9-s + (13.4 + 23.2i)11-s + (−28.1 + 5.67i)12-s + (−16.8 − 9.74i)13-s + (26.8 − 46.5i)14-s + (−5.45 − 9.44i)16-s + 29.1i·17-s + (16.3 + 39.0i)18-s − 47.1·19-s + ⋯ |
L(s) = 1 | + (0.480 − 0.277i)2-s + (0.660 + 0.750i)3-s + (−0.345 + 0.599i)4-s + (0.526 + 0.177i)6-s + (1.60 − 0.924i)7-s + 0.939i·8-s + (−0.126 + 0.991i)9-s + (0.368 + 0.638i)11-s + (−0.678 + 0.136i)12-s + (−0.360 − 0.207i)13-s + (0.513 − 0.888i)14-s + (−0.0852 − 0.147i)16-s + 0.415i·17-s + (0.214 + 0.511i)18-s − 0.568·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.45487 + 1.59970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.45487 + 1.59970i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.43 - 3.89i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.35 + 0.785i)T + (4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-29.6 + 17.1i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-13.4 - 23.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (16.8 + 9.74i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 29.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 47.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-98.0 - 56.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-40.6 - 70.3i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-5.41 + 9.38i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 410. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (-221. + 384. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (294. - 169. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-204. + 118. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 609. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-7.77 + 13.4i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (187. + 108. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 65.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 711. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (478. + 829. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-452. + 261. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-694. + 400. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.89765093132959927328641834674, −11.03630161802186257880401906462, −10.18778047990536082484067930631, −8.882099685874057649655413007551, −8.108076983108089097241352217512, −7.29471470713542207427117799975, −5.01044741096115573360213129387, −4.51183352418844689443368041377, −3.46364875454623504734477135785, −1.90459621216968213566272293266,
1.09599769878849933880774616818, 2.44492135156350127139882265890, 4.26735050674127407583321017626, 5.38241331785593676010811664058, 6.38850005175548359462201446653, 7.62654948469075244342542667645, 8.691843982545287911856799108383, 9.261680709922883048362975016999, 10.84261834397714901138214845023, 11.80296493938962689796996199888