L(s) = 1 | + (0.904 − 0.425i)2-s + (0.481 + 0.876i)3-s + (0.637 − 0.770i)4-s + (−0.425 + 0.904i)5-s + (0.809 + 0.587i)6-s + (−0.684 + 0.728i)7-s + (0.248 − 0.968i)8-s + (−0.535 + 0.844i)9-s + i·10-s + (0.844 + 0.535i)11-s + (0.982 + 0.187i)12-s + (−0.728 + 0.684i)13-s + (−0.309 + 0.951i)14-s + (−0.998 + 0.0627i)15-s + (−0.187 − 0.982i)16-s + (0.809 − 0.587i)17-s + ⋯ |
L(s) = 1 | + (0.904 − 0.425i)2-s + (0.481 + 0.876i)3-s + (0.637 − 0.770i)4-s + (−0.425 + 0.904i)5-s + (0.809 + 0.587i)6-s + (−0.684 + 0.728i)7-s + (0.248 − 0.968i)8-s + (−0.535 + 0.844i)9-s + i·10-s + (0.844 + 0.535i)11-s + (0.982 + 0.187i)12-s + (−0.728 + 0.684i)13-s + (−0.309 + 0.951i)14-s + (−0.998 + 0.0627i)15-s + (−0.187 − 0.982i)16-s + (0.809 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.228535367 + 1.778073470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.228535367 + 1.778073470i\) |
\(L(1)\) |
\(\approx\) |
\(1.759210893 + 0.5698041868i\) |
\(L(1)\) |
\(\approx\) |
\(1.759210893 + 0.5698041868i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 101 | \( 1 \) |
good | 2 | \( 1 + (0.904 - 0.425i)T \) |
| 3 | \( 1 + (0.481 + 0.876i)T \) |
| 5 | \( 1 + (-0.425 + 0.904i)T \) |
| 7 | \( 1 + (-0.684 + 0.728i)T \) |
| 11 | \( 1 + (0.844 + 0.535i)T \) |
| 13 | \( 1 + (-0.728 + 0.684i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.187 + 0.982i)T \) |
| 23 | \( 1 + (0.992 - 0.125i)T \) |
| 29 | \( 1 + (-0.684 - 0.728i)T \) |
| 31 | \( 1 + (0.728 + 0.684i)T \) |
| 37 | \( 1 + (0.876 + 0.481i)T \) |
| 41 | \( 1 + (0.587 - 0.809i)T \) |
| 43 | \( 1 + (0.929 + 0.368i)T \) |
| 47 | \( 1 + (0.929 - 0.368i)T \) |
| 53 | \( 1 + (-0.770 + 0.637i)T \) |
| 59 | \( 1 + (0.982 - 0.187i)T \) |
| 61 | \( 1 + (-0.770 - 0.637i)T \) |
| 67 | \( 1 + (-0.481 + 0.876i)T \) |
| 71 | \( 1 + (0.876 - 0.481i)T \) |
| 73 | \( 1 + (-0.125 - 0.992i)T \) |
| 79 | \( 1 + (-0.992 - 0.125i)T \) |
| 83 | \( 1 + (0.125 - 0.992i)T \) |
| 89 | \( 1 + (-0.982 - 0.187i)T \) |
| 97 | \( 1 + (-0.637 + 0.770i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.83493267859123098272933500747, −28.79088005793365649699504659625, −27.14598930742236220716845142191, −25.931559685470528743790693368020, −25.00657207958926224171508684552, −24.18286503876605983000526170530, −23.44632330335550872050561948183, −22.41200910592022484366466143281, −20.95875195961935709496503583387, −19.88229605299304904752176439887, −19.37316409620248288103264099296, −17.28871520040324560857051572281, −16.70547098727099125858073784750, −15.295474580034476002128260897198, −14.2131571822693774252671168735, −13.043676234566917304765630728943, −12.59990511679218483599016795218, −11.32464096349101608260480342789, −9.19397184661163302542163651797, −7.92943424144806102795705105444, −7.02695704731430709825402847833, −5.74914794342959337182184137028, −4.12798457334424874997565416566, −2.97535055663525170583178888169, −0.91641772686170193005340052930,
2.38201092341138310130477929305, 3.378403116852398987774734950830, 4.480253019327085016352025983, 5.99351235687147116666527907436, 7.318317868241039296277970751923, 9.33985317479010925252163586949, 10.146052269246992763264489820353, 11.471362823645664933619956525796, 12.36455259988845340648286507282, 14.08036185839787387741714367811, 14.73820661441871424505374097754, 15.5550158296065096115189588504, 16.68090420727135068567359921445, 18.931508627490711928817386192939, 19.35015563150456951764587713840, 20.631844629466327745588782212965, 21.6686986929377619635011646930, 22.48558328962824166270685687243, 23.06363833339370133932776516533, 24.86131251338332590275072518826, 25.55769403485135841185351614996, 26.89403477595449135250077633684, 27.81116947034866722985779331121, 28.95173959384375630567876573734, 30.08476600761446484701057691353