L(s) = 1 | + (0.945 − 0.324i)2-s + (−0.401 − 0.915i)3-s + (0.789 − 0.614i)4-s + (0.245 − 0.969i)5-s + (−0.677 − 0.735i)6-s + (0.789 + 0.614i)7-s + (0.546 − 0.837i)8-s + (−0.677 + 0.735i)9-s + (−0.0825 − 0.996i)10-s + (−0.677 − 0.735i)11-s + (−0.879 − 0.475i)12-s + (−0.401 − 0.915i)13-s + (0.945 + 0.324i)14-s + (−0.986 + 0.164i)15-s + (0.245 − 0.969i)16-s + (0.789 + 0.614i)17-s + ⋯ |
L(s) = 1 | + (0.945 − 0.324i)2-s + (−0.401 − 0.915i)3-s + (0.789 − 0.614i)4-s + (0.245 − 0.969i)5-s + (−0.677 − 0.735i)6-s + (0.789 + 0.614i)7-s + (0.546 − 0.837i)8-s + (−0.677 + 0.735i)9-s + (−0.0825 − 0.996i)10-s + (−0.677 − 0.735i)11-s + (−0.879 − 0.475i)12-s + (−0.401 − 0.915i)13-s + (0.945 + 0.324i)14-s + (−0.986 + 0.164i)15-s + (0.245 − 0.969i)16-s + (0.789 + 0.614i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9573099623 - 1.905395156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9573099623 - 1.905395156i\) |
\(L(1)\) |
\(\approx\) |
\(1.314097321 - 1.085485995i\) |
\(L(1)\) |
\(\approx\) |
\(1.314097321 - 1.085485995i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 + (0.945 - 0.324i)T \) |
| 3 | \( 1 + (-0.401 - 0.915i)T \) |
| 5 | \( 1 + (0.245 - 0.969i)T \) |
| 7 | \( 1 + (0.789 + 0.614i)T \) |
| 11 | \( 1 + (-0.677 - 0.735i)T \) |
| 13 | \( 1 + (-0.401 - 0.915i)T \) |
| 17 | \( 1 + (0.789 + 0.614i)T \) |
| 23 | \( 1 + (-0.401 + 0.915i)T \) |
| 29 | \( 1 + (0.789 + 0.614i)T \) |
| 31 | \( 1 + (0.945 + 0.324i)T \) |
| 37 | \( 1 + (-0.677 - 0.735i)T \) |
| 41 | \( 1 + (-0.986 + 0.164i)T \) |
| 43 | \( 1 + (-0.0825 + 0.996i)T \) |
| 47 | \( 1 + (-0.677 - 0.735i)T \) |
| 53 | \( 1 + (-0.677 - 0.735i)T \) |
| 59 | \( 1 + (-0.986 + 0.164i)T \) |
| 61 | \( 1 + (0.546 + 0.837i)T \) |
| 67 | \( 1 + (0.546 - 0.837i)T \) |
| 71 | \( 1 + (0.546 - 0.837i)T \) |
| 73 | \( 1 + (0.789 + 0.614i)T \) |
| 79 | \( 1 + (-0.0825 + 0.996i)T \) |
| 83 | \( 1 + (0.245 + 0.969i)T \) |
| 89 | \( 1 + (0.789 - 0.614i)T \) |
| 97 | \( 1 + (0.546 + 0.837i)T \) |
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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
−24.99668018856508961100436633972, −23.77728671738657854528672410111, −23.18243699242862122663903917249, −22.46712920446869042548220546861, −21.6216885866618744571932385962, −20.889022096088349752589257718796, −20.32126377720278905879902406324, −18.76377067379077245437361699689, −17.55098434868021097977549585249, −16.99366393397615846599844304661, −15.88929797582037706038607037588, −15.11161538312784284840263836171, −14.2858856083950229913753569001, −13.81057542171986315570640034905, −12.168380230071949486299816154089, −11.50917304714069221694889286988, −10.51546923525657629965653540647, −9.89630433777863898400767582694, −8.1540961848407321005624953332, −7.12015045031898150756544787160, −6.26998088387339972512619255432, −5.00291915222878479239122035990, −4.45034971620076127760393275104, −3.26197482875129472160650779504, −2.15447390044491569301603856565,
1.06084284250996130059423979455, 2.01646290345405726885886219072, 3.211353360591242591008730926251, 5.020118943825468379246642297481, 5.35931616546937660021590032155, 6.256286271800939526002978173898, 7.7851378744670702807169185791, 8.38440890415317316928684204194, 10.06088517195755382507139769726, 11.09996916491519455340647024700, 12.0749529222131765225626666855, 12.56246269834356708379229878961, 13.45232146002272314318466864285, 14.212143986095941220191592863314, 15.39135955563022943259065100918, 16.33069052829049575876551483288, 17.37758551761727151060883548567, 18.23555083999707234695838611114, 19.33010223741586292060536261385, 20.03167377886204972672599515843, 21.2316227384842700271962015839, 21.56216512946603990558880214909, 22.82036189616012350613009007396, 23.71234915580540899392892660904, 24.22097794081637696254388440567