L(s) = 1 | + (−0.994 − 0.101i)2-s + (−0.994 + 0.101i)3-s + (0.979 + 0.201i)4-s + (0.688 − 0.724i)5-s + 6-s + (0.347 + 0.937i)7-s + (−0.954 − 0.299i)8-s + (0.979 − 0.201i)9-s + (−0.758 + 0.651i)10-s + (0.0506 − 0.998i)11-s + (−0.994 − 0.101i)12-s + (0.979 − 0.201i)13-s + (−0.250 − 0.968i)14-s + (−0.612 + 0.790i)15-s + (0.918 + 0.394i)16-s + (0.758 − 0.651i)17-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.101i)2-s + (−0.994 + 0.101i)3-s + (0.979 + 0.201i)4-s + (0.688 − 0.724i)5-s + 6-s + (0.347 + 0.937i)7-s + (−0.954 − 0.299i)8-s + (0.979 − 0.201i)9-s + (−0.758 + 0.651i)10-s + (0.0506 − 0.998i)11-s + (−0.994 − 0.101i)12-s + (0.979 − 0.201i)13-s + (−0.250 − 0.968i)14-s + (−0.612 + 0.790i)15-s + (0.918 + 0.394i)16-s + (0.758 − 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9048540990 - 0.6122517454i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9048540990 - 0.6122517454i\) |
\(L(1)\) |
\(\approx\) |
\(0.6856489059 - 0.1475121719i\) |
\(L(1)\) |
\(\approx\) |
\(0.6856489059 - 0.1475121719i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (-0.994 - 0.101i)T \) |
| 3 | \( 1 + (-0.994 + 0.101i)T \) |
| 5 | \( 1 + (0.688 - 0.724i)T \) |
| 7 | \( 1 + (0.347 + 0.937i)T \) |
| 11 | \( 1 + (0.0506 - 0.998i)T \) |
| 13 | \( 1 + (0.979 - 0.201i)T \) |
| 17 | \( 1 + (0.758 - 0.651i)T \) |
| 19 | \( 1 + (-0.688 - 0.724i)T \) |
| 23 | \( 1 + (0.954 + 0.299i)T \) |
| 29 | \( 1 + (0.250 - 0.968i)T \) |
| 31 | \( 1 + (0.758 + 0.651i)T \) |
| 37 | \( 1 + (-0.347 + 0.937i)T \) |
| 41 | \( 1 + (-0.528 + 0.848i)T \) |
| 43 | \( 1 + (-0.347 - 0.937i)T \) |
| 47 | \( 1 + (-0.250 + 0.968i)T \) |
| 53 | \( 1 + (0.347 + 0.937i)T \) |
| 59 | \( 1 + (-0.347 - 0.937i)T \) |
| 61 | \( 1 + (-0.688 - 0.724i)T \) |
| 67 | \( 1 + (0.528 + 0.848i)T \) |
| 71 | \( 1 + (0.874 - 0.485i)T \) |
| 73 | \( 1 + (0.820 + 0.571i)T \) |
| 79 | \( 1 + (0.528 - 0.848i)T \) |
| 83 | \( 1 + (-0.612 - 0.790i)T \) |
| 89 | \( 1 + (0.347 - 0.937i)T \) |
| 97 | \( 1 + (0.874 - 0.485i)T \) |
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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
−25.45712741730670603527655096991, −24.36157561930511649028755696445, −23.26009992421294527647227954172, −22.87770659183374823101488856310, −21.25466444023285590590781136584, −20.93606970159771826839552501498, −19.53045255847156906337174496091, −18.5141239011620962236647306107, −17.960957028822274325818368308205, −17.05841909712155972094826641508, −16.65212648430057847418612890541, −15.301813542701793591253474316206, −14.42660706645037872627442618211, −13.08943175785547970379776947453, −11.97812704769027593636278194901, −10.70360787038104523469159910426, −10.579509966184468261213047190909, −9.56629578453340206357505422628, −8.072701236674001274022481375051, −6.983335360341502430044435116404, −6.467299506478768289034470391703, −5.32560235758411967417514665119, −3.763154165942292378462161115006, −1.938009513634569521271537607630, −1.09593720648970727789014245202,
0.63114450444946619071710485211, 1.51876132597866513190399591966, 3.01639448038303343184090965470, 4.919153466983595250563777648266, 5.85582449783903192534817642444, 6.543171858925352657045121368603, 8.16542158855448447263649646943, 8.91802214641144724625960682139, 9.843146428155982696323610133484, 10.93614319939657073018527048395, 11.647139875211704991151913242704, 12.53465822848824997716746600767, 13.62596227394073504074062019695, 15.352574281756787740414305251751, 15.991816030368337128985752560949, 16.93495765782660380209450065431, 17.507773576980350426264489002317, 18.49155832269946370191447901227, 19.01567526500630559624050198607, 20.559036090546870330825827817451, 21.346930352542665044630803294472, 21.69843909814459726265376864079, 23.21634465242036312765870901892, 24.21566226567802179651194333637, 24.91106051009643791919274780901