L(s) = 1 | + (0.239 − 0.970i)2-s + (−0.822 + 0.568i)3-s + (−0.885 − 0.464i)4-s + (0.935 + 0.354i)5-s + (0.354 + 0.935i)6-s + (0.970 + 0.239i)7-s + (−0.663 + 0.748i)8-s + (0.354 − 0.935i)9-s + (0.568 − 0.822i)10-s + (−0.120 − 0.992i)11-s + (0.992 − 0.120i)12-s + (0.885 − 0.464i)13-s + (0.464 − 0.885i)14-s + (−0.970 + 0.239i)15-s + (0.568 + 0.822i)16-s + (0.748 − 0.663i)17-s + ⋯ |
L(s) = 1 | + (0.239 − 0.970i)2-s + (−0.822 + 0.568i)3-s + (−0.885 − 0.464i)4-s + (0.935 + 0.354i)5-s + (0.354 + 0.935i)6-s + (0.970 + 0.239i)7-s + (−0.663 + 0.748i)8-s + (0.354 − 0.935i)9-s + (0.568 − 0.822i)10-s + (−0.120 − 0.992i)11-s + (0.992 − 0.120i)12-s + (0.885 − 0.464i)13-s + (0.464 − 0.885i)14-s + (−0.970 + 0.239i)15-s + (0.568 + 0.822i)16-s + (0.748 − 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.398560573 - 0.7501349830i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.398560573 - 0.7501349830i\) |
\(L(1)\) |
\(\approx\) |
\(1.090759177 - 0.4053417945i\) |
\(L(1)\) |
\(\approx\) |
\(1.090759177 - 0.4053417945i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (0.239 - 0.970i)T \) |
| 3 | \( 1 + (-0.822 + 0.568i)T \) |
| 5 | \( 1 + (0.935 + 0.354i)T \) |
| 7 | \( 1 + (0.970 + 0.239i)T \) |
| 11 | \( 1 + (-0.120 - 0.992i)T \) |
| 13 | \( 1 + (0.885 - 0.464i)T \) |
| 17 | \( 1 + (0.748 - 0.663i)T \) |
| 19 | \( 1 + (0.464 + 0.885i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.120 + 0.992i)T \) |
| 31 | \( 1 + (0.992 + 0.120i)T \) |
| 37 | \( 1 + (-0.568 - 0.822i)T \) |
| 41 | \( 1 + (-0.992 + 0.120i)T \) |
| 43 | \( 1 + (-0.568 + 0.822i)T \) |
| 47 | \( 1 + (-0.354 - 0.935i)T \) |
| 59 | \( 1 + (0.354 + 0.935i)T \) |
| 61 | \( 1 + (-0.663 + 0.748i)T \) |
| 67 | \( 1 + (0.464 - 0.885i)T \) |
| 71 | \( 1 + (-0.822 - 0.568i)T \) |
| 73 | \( 1 + (-0.663 - 0.748i)T \) |
| 79 | \( 1 + (-0.239 - 0.970i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.748 + 0.663i)T \) |
| 97 | \( 1 + (-0.354 + 0.935i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−33.55207663200416916636215093603, −32.55623809369897480580419321077, −30.82459814742208857890742216664, −30.15265392442899208850398115516, −28.48525074697870423265404110159, −27.74792287281963803306730704109, −26.04777150545499059747345866946, −25.057164586170032449943043552173, −24.01514807141779943296253049438, −23.26948069515502870651141613557, −21.916211972769956501977173690441, −20.87295461584376817575197906610, −18.62551720062713958897929827158, −17.50051314275075842440379889700, −17.151584940513792919363116201896, −15.57906507224658092649513815504, −13.98593796155090179094590997585, −13.149243160522752551630059547666, −11.75744007193267015942985928047, −9.95587358768575709386983605357, −8.24600665569764319278223331772, −6.92530285809496626496719740886, −5.65406282516164545260130057284, −4.6309877172181488878369887792, −1.44809098712415726253580982193,
1.21994382139713066928852310299, 3.261331892954154823875513271152, 5.08246361335702032640867124389, 5.95607712933566577830170512500, 8.678785342995819310064487266501, 10.168166043398435789246541886955, 10.9556109890602129120740152741, 12.0843170029523656104740108970, 13.68228936155280571313421794058, 14.74041893653025115070918738764, 16.55059948064708577237297470635, 18.01978191374784636148654167675, 18.48435611115395443761293641375, 20.73838340860255807829048362772, 21.20331756903097404158551366063, 22.26385734431450532880970499032, 23.2595327444424705059061441090, 24.66300850673611152615687616145, 26.5461071020070558405387409475, 27.48090975656765218575145811758, 28.482948514530618004847165343255, 29.4916506569730252538842023592, 30.286735258768792959915058315766, 31.784576983038491792436657611342, 32.91231128883179472603724916890