L(s) = 1 | + (0.850 − 0.526i)2-s + (−0.361 − 0.932i)3-s + (0.445 − 0.895i)4-s + (0.526 − 0.850i)5-s + (−0.798 − 0.602i)6-s + (0.982 + 0.183i)7-s + (−0.0922 − 0.995i)8-s + (−0.739 + 0.673i)9-s − i·10-s + (−0.445 + 0.895i)11-s + (−0.995 − 0.0922i)12-s + (−0.183 + 0.982i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (−0.0922 + 0.995i)17-s + ⋯ |
L(s) = 1 | + (0.850 − 0.526i)2-s + (−0.361 − 0.932i)3-s + (0.445 − 0.895i)4-s + (0.526 − 0.850i)5-s + (−0.798 − 0.602i)6-s + (0.982 + 0.183i)7-s + (−0.0922 − 0.995i)8-s + (−0.739 + 0.673i)9-s − i·10-s + (−0.445 + 0.895i)11-s + (−0.995 − 0.0922i)12-s + (−0.183 + 0.982i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (−0.0922 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9521801630 - 1.403251049i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9521801630 - 1.403251049i\) |
\(L(1)\) |
\(\approx\) |
\(1.229636932 - 0.9930239716i\) |
\(L(1)\) |
\(\approx\) |
\(1.229636932 - 0.9930239716i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (0.850 - 0.526i)T \) |
| 3 | \( 1 + (-0.361 - 0.932i)T \) |
| 5 | \( 1 + (0.526 - 0.850i)T \) |
| 7 | \( 1 + (0.982 + 0.183i)T \) |
| 11 | \( 1 + (-0.445 + 0.895i)T \) |
| 13 | \( 1 + (-0.183 + 0.982i)T \) |
| 17 | \( 1 + (-0.0922 + 0.995i)T \) |
| 19 | \( 1 + (0.273 - 0.961i)T \) |
| 23 | \( 1 + (-0.798 + 0.602i)T \) |
| 29 | \( 1 + (0.798 - 0.602i)T \) |
| 31 | \( 1 + (-0.961 + 0.273i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.961 + 0.273i)T \) |
| 47 | \( 1 + (0.673 + 0.739i)T \) |
| 53 | \( 1 + (-0.961 - 0.273i)T \) |
| 59 | \( 1 + (0.739 - 0.673i)T \) |
| 61 | \( 1 + (-0.739 - 0.673i)T \) |
| 67 | \( 1 + (0.183 - 0.982i)T \) |
| 71 | \( 1 + (0.895 - 0.445i)T \) |
| 73 | \( 1 + (-0.982 + 0.183i)T \) |
| 79 | \( 1 + (0.361 - 0.932i)T \) |
| 83 | \( 1 + (0.995 - 0.0922i)T \) |
| 89 | \( 1 + (-0.526 + 0.850i)T \) |
| 97 | \( 1 + (-0.895 - 0.445i)T \) |
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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
−29.159411425849633874050223375201, −27.37072766011926037552674046917, −26.84095324108613319554730877413, −25.82581271826247622203254921839, −24.77985983302902043906743742472, −23.67530756585596811125782520712, −22.60326714742755543933682359593, −22.03405561803539736877328472761, −21.026350260392019885873398278607, −20.40665308301178096966941785790, −18.271036696212974001617325912573, −17.49732309961956495360508652549, −16.36936501000410765722305575468, −15.4411120318088265534714682596, −14.38408598661748142853166683493, −13.88079340236350843396163366084, −12.16877828722364141237800424198, −11.03764112163629837347120029753, −10.336744631069305753924854027299, −8.57182481715359151323586062904, −7.33637273374736435522590455758, −5.8283565982829672949262753987, −5.23273368855771772135354029560, −3.78205935404568490393340074591, −2.652136213025971187811837318745,
1.52379829179776721008515893705, 2.20290425511755926539316669681, 4.49349097728435088359663540741, 5.29631171984679554711861749994, 6.46236930427074898485285901349, 7.83370020451553174944554503356, 9.32001163447182937950411811648, 10.79655349979045015926415671503, 11.889121711773314227838693947829, 12.58815992082886195154141933748, 13.585359592721407915068990240718, 14.4216186522774822842730760216, 15.81277863931557009162468839238, 17.28650719143751450979521800915, 18.008944049158013630455258446732, 19.33485127014140252896511788838, 20.2465921923360333156483521826, 21.27231945639354400022623286046, 22.05992319726049213345841118897, 23.59227335528128956473610527505, 23.92951773200729867831848294490, 24.77421295589153702485408730520, 25.827699056534861714487966312, 27.87011119680052609985306596265, 28.43589590684128869098372452164