L(s) = 1 | + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2344106996 - 1.783477542i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2344106996 - 1.783477542i\) |
\(L(1)\) |
\(\approx\) |
\(1.083274928 - 0.9285580541i\) |
\(L(1)\) |
\(\approx\) |
\(1.083274928 - 0.9285580541i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−18.62369980421418805972925375248, −18.21472203139118263776989753084, −17.38347196067864875236012162076, −16.70518479880121096236344099580, −15.6851018625825318616413287276, −15.41782871223048292190977166549, −14.70342962423466226752735338440, −14.41513362461366503750967326815, −13.5586664818003772654447009133, −12.73101085733410971988627360130, −12.220701666039358768887117177225, −11.572173437805317327985014005638, −10.44955367043809241105465451738, −9.66307825431775313027163798515, −8.81754005231109667032835574391, −8.03683817394516963387037566831, −7.74761353216688026464629810573, −7.25101029487001388804564240504, −6.34984800768877230990634910331, −5.00371515932131302143626019229, −4.70185828848972007683518294571, −4.050921918230193196269399248112, −3.13191496894794255929095529783, −2.47190332640152726037355974440, −1.366300720955424583676482217546,
0.35323061138685853942421875367, 1.443510711230486274667723555548, 2.21943655263246725925594880837, 3.16587416557300189880339112526, 3.461240400327381657013443279250, 4.59585747864652519730251282924, 4.837221335511266988462415540366, 5.86920914013898004384150077353, 7.10805909629457152783756466468, 7.86056232325833723314290183231, 8.29790399809042192902486626601, 9.03592784598842115202214046743, 10.07825429069116346228077108788, 10.37755171936605636025998989071, 11.46195225189112426679693373950, 11.9849751024705106743982167034, 12.381133822479941107114935041273, 13.57826987827587196619685697535, 13.917036067959282272476987391753, 14.577681404820744147689934191232, 15.11515508865017971052463341871, 15.84077296780384883814848457273, 16.51226627954493130747715307778, 17.80457369105458399118882772866, 18.41886349287692665033510556422