L(s) = 1 | + (0.858 − 0.512i)2-s + (−0.393 + 0.919i)3-s + (0.473 − 0.880i)4-s + (−0.393 − 0.919i)5-s + (0.134 + 0.990i)6-s + (0.858 + 0.512i)7-s + (−0.0448 − 0.998i)8-s + (−0.691 − 0.722i)9-s + (−0.809 − 0.587i)10-s + (0.753 + 0.657i)11-s + (0.623 + 0.781i)12-s + (−0.550 − 0.834i)13-s + 14-s + 15-s + (−0.550 − 0.834i)16-s + (0.936 − 0.351i)17-s + ⋯ |
L(s) = 1 | + (0.858 − 0.512i)2-s + (−0.393 + 0.919i)3-s + (0.473 − 0.880i)4-s + (−0.393 − 0.919i)5-s + (0.134 + 0.990i)6-s + (0.858 + 0.512i)7-s + (−0.0448 − 0.998i)8-s + (−0.691 − 0.722i)9-s + (−0.809 − 0.587i)10-s + (0.753 + 0.657i)11-s + (0.623 + 0.781i)12-s + (−0.550 − 0.834i)13-s + 14-s + 15-s + (−0.550 − 0.834i)16-s + (0.936 − 0.351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.572223143 - 0.6800293129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.572223143 - 0.6800293129i\) |
\(L(1)\) |
\(\approx\) |
\(1.459932723 - 0.3715353860i\) |
\(L(1)\) |
\(\approx\) |
\(1.459932723 - 0.3715353860i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 211 | \( 1 \) |
good | 2 | \( 1 + (0.858 - 0.512i)T \) |
| 3 | \( 1 + (-0.393 + 0.919i)T \) |
| 5 | \( 1 + (-0.393 - 0.919i)T \) |
| 7 | \( 1 + (0.858 + 0.512i)T \) |
| 11 | \( 1 + (0.753 + 0.657i)T \) |
| 13 | \( 1 + (-0.550 - 0.834i)T \) |
| 17 | \( 1 + (0.936 - 0.351i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.309 + 0.951i)T \) |
| 29 | \( 1 + (-0.550 - 0.834i)T \) |
| 31 | \( 1 + (0.623 + 0.781i)T \) |
| 37 | \( 1 + (0.753 - 0.657i)T \) |
| 41 | \( 1 + (-0.963 - 0.266i)T \) |
| 43 | \( 1 + (0.623 + 0.781i)T \) |
| 47 | \( 1 + (-0.691 + 0.722i)T \) |
| 53 | \( 1 + (0.473 + 0.880i)T \) |
| 59 | \( 1 + (-0.963 - 0.266i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.222 + 0.974i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.900 + 0.433i)T \) |
| 79 | \( 1 + (-0.0448 + 0.998i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.995 - 0.0896i)T \) |
| 97 | \( 1 + (0.983 + 0.178i)T \) |
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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
−26.678935384761191271532704496412, −25.661232164228568092872659680319, −24.56103947747966422427500384020, −23.998889945574270711764581868079, −23.167218330183320165228982706324, −22.39952221584918861226616781387, −21.50602777444543802814284537132, −20.254941345511130155541011564594, −19.075027903104315470678281883294, −18.2645781773780787892600083252, −16.94107789370493663458374679055, −16.592816908993516631593087579081, −14.72680574155312298746375953790, −14.39492123213288864969743456946, −13.49093433410153675898429128336, −12.02999549993471815440235132956, −11.63214698019322300418089329623, −10.56838402494797193718274448162, −8.36773074980155695919319971075, −7.52938701864582841479529662005, −6.73391338666329167073267752537, −5.796507883019477265686672048781, −4.421066469920091850487159563446, −3.19763360939598405122728485782, −1.732180065001980564568820930309,
1.22171054872651388135143513509, 2.97918042549253212231088157870, 4.308873031292565558807388916686, 4.99659189785895252391550433672, 5.7608913511465431605695559551, 7.54901904371055182393704544220, 9.10861945607653871105113138544, 9.88117919583357858216613478116, 11.26909821989098285895399831425, 11.861499804471314965519599321721, 12.66871765884789513194284593224, 14.1383594521851267613054993650, 15.14379716683766874279920490527, 15.62954272772313995524054541194, 16.91484119721912712913139026476, 17.81894730555174865601596152801, 19.49060449776072081952238351516, 20.256560830739088021624674840079, 21.02205270999706394201554799561, 21.75566605127409212162380275356, 22.7482679448235457847492156674, 23.49711607574758371431091575056, 24.57285645313346658942010154666, 25.2584662260319906098493684967, 27.053553635890097764693602767619