L(s) = 1 | + (−0.959 − 0.281i)3-s + (0.142 − 0.989i)7-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.654 − 0.755i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.654 − 0.755i)27-s + (0.654 − 0.755i)29-s + (−0.959 + 0.281i)31-s + (0.142 + 0.989i)33-s + (0.841 + 0.540i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)3-s + (0.142 − 0.989i)7-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.654 − 0.755i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.654 − 0.755i)27-s + (0.654 − 0.755i)29-s + (−0.959 + 0.281i)31-s + (0.142 + 0.989i)33-s + (0.841 + 0.540i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3402918906 - 0.7869081826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3402918906 - 0.7869081826i\) |
\(L(1)\) |
\(\approx\) |
\(0.7172701756 - 0.3161823947i\) |
\(L(1)\) |
\(\approx\) |
\(0.7172701756 - 0.3161823947i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 11 | \( 1 + (-0.415 - 0.909i)T \) |
| 13 | \( 1 + (-0.142 - 0.989i)T \) |
| 17 | \( 1 + (0.654 - 0.755i)T \) |
| 19 | \( 1 + (0.654 + 0.755i)T \) |
| 29 | \( 1 + (0.654 - 0.755i)T \) |
| 31 | \( 1 + (-0.959 + 0.281i)T \) |
| 37 | \( 1 + (0.841 + 0.540i)T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (-0.959 - 0.281i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.142 + 0.989i)T \) |
| 59 | \( 1 + (0.142 + 0.989i)T \) |
| 61 | \( 1 + (0.959 - 0.281i)T \) |
| 67 | \( 1 + (0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.654 + 0.755i)T \) |
| 79 | \( 1 + (-0.142 - 0.989i)T \) |
| 83 | \( 1 + (0.841 + 0.540i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
| 97 | \( 1 + (-0.841 + 0.540i)T \) |
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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
−21.97119467235695433668868703954, −21.61903851397928666527841225768, −20.85616931022765015289868102475, −19.78025561786993413736654125701, −18.821464167453847979740478910484, −18.06530674712528458253802156042, −17.59788893327935582941788865348, −16.47815144131779457391339215124, −16.00051406443508859289306985526, −15.02147235803011566755018623112, −14.48804562067106626029004433670, −12.993016869117587340416446498227, −12.47968641764022681505121720329, −11.6170899037978891877669016401, −11.05003494766414635824985310100, −9.84808493735841422753256560439, −9.42188841172799598889537520769, −8.241268879492960813466585374934, −7.13698223337944854995652660684, −6.377096882744631572650343850188, −5.35163404731468863668698322207, −4.83232877970868206823586609518, −3.74983262458940939329217518814, −2.40611465316609948576009554926, −1.38242164620235641520976286816,
0.48291689475437093551157035011, 1.324196285767543594802813905109, 2.88653473592512692798010176796, 3.89213901251450217216610176684, 5.06040312926030342869213396057, 5.6495417791990893728878285258, 6.63151647227112419484795662781, 7.64525576482141535298424874578, 8.05136494689100988588594589690, 9.66103606253990615429828449858, 10.35953646026294348483764836652, 11.038672273778003424541475566969, 11.83623798492702426728106463300, 12.72717787792277283364314608845, 13.528519743401969253263266348, 14.17407986540850327250634688504, 15.39651856838968666916856826426, 16.39631805426593074330990746989, 16.65167252475799409246276465124, 17.72274762126471234562459417101, 18.26132832632774707353567535074, 19.08984915239012951428734656794, 20.05777189538174479579448591714, 20.841987721832339185540832194498, 21.65399201778978756140816578992