L(s) = 1 | + (0.281 + 0.959i)3-s + (−0.989 + 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.989 − 0.142i)13-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (−0.755 − 0.654i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (0.989 + 0.142i)33-s + (−0.540 − 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯ |
L(s) = 1 | + (0.281 + 0.959i)3-s + (−0.989 + 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.989 − 0.142i)13-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (−0.755 − 0.654i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (0.989 + 0.142i)33-s + (−0.540 − 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.180322804 - 0.1559568647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.180322804 - 0.1559568647i\) |
\(L(1)\) |
\(\approx\) |
\(0.9878009539 + 0.1365373563i\) |
\(L(1)\) |
\(\approx\) |
\(0.9878009539 + 0.1365373563i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.281 + 0.959i)T \) |
| 7 | \( 1 + (-0.989 + 0.142i)T \) |
| 11 | \( 1 + (0.415 - 0.909i)T \) |
| 13 | \( 1 + (-0.989 - 0.142i)T \) |
| 17 | \( 1 + (0.755 - 0.654i)T \) |
| 19 | \( 1 + (0.654 - 0.755i)T \) |
| 29 | \( 1 + (-0.654 - 0.755i)T \) |
| 31 | \( 1 + (0.959 + 0.281i)T \) |
| 37 | \( 1 + (-0.540 - 0.841i)T \) |
| 41 | \( 1 + (0.841 + 0.540i)T \) |
| 43 | \( 1 + (0.281 + 0.959i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.989 - 0.142i)T \) |
| 59 | \( 1 + (0.142 - 0.989i)T \) |
| 61 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (-0.909 + 0.415i)T \) |
| 71 | \( 1 + (-0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.755 + 0.654i)T \) |
| 79 | \( 1 + (-0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.540 - 0.841i)T \) |
| 89 | \( 1 + (0.959 - 0.281i)T \) |
| 97 | \( 1 + (-0.540 + 0.841i)T \) |
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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
−22.26881160606484570748949210378, −20.88352845632638128703651313016, −20.20369524603110492628488690724, −19.36552932546360948689498315871, −19.01435529685952857930820801570, −18.01302273409350860684743726601, −17.16061767121994219336110529711, −16.61388663780389560074186833526, −15.402965219076022090447722442926, −14.59049168367465112544014147194, −13.926753460084915301377066352395, −12.93708719712483817470309566967, −12.307733529582039506045212611094, −11.84995506117387568640960011694, −10.35338026638543244074079815377, −9.67722888478795112477365110544, −8.84512893213599703161548242913, −7.656406600133449918214436752569, −7.17336668584554714458751498271, −6.29277783260674745472454775899, −5.41878890863805973016113682693, −4.040048462735896370336423031057, −3.1201063681485783213254000608, −2.148770593823986418809291079136, −1.09447823300471301610536984717,
0.57297791721277454067446080164, 2.536577261070399776776652032983, 3.143230025703290309003557445399, 4.03312037427022717325291584614, 5.147068261861576991838262631726, 5.847842343689723832217315563857, 6.974785099226948757370269872359, 7.97840774599258458173261689602, 9.04593531435105658926042292407, 9.5967869099215552004849742591, 10.2533811253047220021042945899, 11.34742415319224545878950697763, 12.013246876299245021021155606757, 13.17390769504835206098894260985, 13.95452903600913125422569986875, 14.696844020228781328581011931265, 15.60947457327138921856771606406, 16.25728416414853697706744156092, 16.82009128155122252702604140993, 17.77279009759668341341951434517, 19.049521009410099796778247126025, 19.482411631804752376773485591655, 20.2262621343167396834357482991, 21.22557332470062119765084317571, 21.79375148525641290428256929474