L(s) = 1 | + (−0.888 + 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (−0.723 + 0.690i)5-s + (0.415 − 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)10-s + (0.888 + 0.458i)11-s + (0.0475 + 0.998i)12-s + (−0.654 − 0.755i)13-s + (0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (0.235 + 0.971i)18-s + (0.995 + 0.0950i)19-s + ⋯ |
L(s) = 1 | + (−0.888 + 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (−0.723 + 0.690i)5-s + (0.415 − 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)10-s + (0.888 + 0.458i)11-s + (0.0475 + 0.998i)12-s + (−0.654 − 0.755i)13-s + (0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (0.235 + 0.971i)18-s + (0.995 + 0.0950i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4302854454 + 0.5757764878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4302854454 + 0.5757764878i\) |
\(L(1)\) |
\(\approx\) |
\(0.4952252589 + 0.2610176103i\) |
\(L(1)\) |
\(\approx\) |
\(0.4952252589 + 0.2610176103i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.888 + 0.458i)T \) |
| 3 | \( 1 + (-0.786 + 0.618i)T \) |
| 5 | \( 1 + (-0.723 + 0.690i)T \) |
| 11 | \( 1 + (0.888 + 0.458i)T \) |
| 13 | \( 1 + (-0.654 - 0.755i)T \) |
| 17 | \( 1 + (0.995 - 0.0950i)T \) |
| 19 | \( 1 + (0.995 + 0.0950i)T \) |
| 29 | \( 1 + (0.415 - 0.909i)T \) |
| 31 | \( 1 + (0.928 - 0.371i)T \) |
| 37 | \( 1 + (-0.235 + 0.971i)T \) |
| 41 | \( 1 + (-0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.142 + 0.989i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.981 - 0.189i)T \) |
| 59 | \( 1 + (-0.327 + 0.945i)T \) |
| 61 | \( 1 + (0.786 + 0.618i)T \) |
| 67 | \( 1 + (-0.0475 + 0.998i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (0.580 - 0.814i)T \) |
| 79 | \( 1 + (-0.981 + 0.189i)T \) |
| 83 | \( 1 + (0.959 - 0.281i)T \) |
| 89 | \( 1 + (-0.928 - 0.371i)T \) |
| 97 | \( 1 + (0.959 + 0.281i)T \) |
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
−27.42961472863411876149757483001, −26.70068020597964829461829362733, −25.139800250855197404283567392975, −24.4769154972730850753994914507, −23.54068282759384053374443227770, −22.28545348937659147047103524505, −21.347259500683415890136222170102, −20.00753446867368133346580980298, −19.30581398783066191816159977931, −18.52055828957267473755751734578, −17.2449486497694171784815692242, −16.66023298929063169319248817840, −15.847245762368566706403171951155, −14.00350109360543516343541674778, −12.49142443827099543569140542026, −11.96963654808623288462179412590, −11.20017791512450687120609729262, −9.81562161147550340133068182138, −8.63238615741259887188192503050, −7.577603754046187670672260828893, −6.64642450298525640798062978156, −5.0236748441712767480061258921, −3.50349297371994567153703018355, −1.61031269589880294188984360822, −0.58222635070738384519545734384,
0.90834441733907707592134052190, 3.11650513427669707604887076300, 4.7031928296368326475880496294, 6.008315374614327060797844564804, 7.04304535277208822868478947937, 8.05166636065196779567551355722, 9.68459761484689493925413974355, 10.20052355710616215338851322923, 11.50328232147222134074970218224, 12.0541193737018072358702249111, 14.36067407625573009235181750161, 15.15802529483726610512006712879, 15.94118037521099555544737543622, 17.00874020723341952833800801985, 17.76078139766810685961425167888, 18.80783349756561796428980991121, 19.79962231226624152754829475474, 20.8131196450841206805251617560, 22.39143946507653326783570797118, 22.86993439155390506055308133012, 23.9662577202722946548864775985, 25.04702857533332194368343386550, 26.190866557268799388767513281083, 27.12084720668276863004482164726, 27.523447190397227817864569438022