L(s) = 1 | − 2.99·5-s − i·7-s − 5.36i·11-s + 4.55i·13-s + 0.958i·17-s − 0.532·19-s + 2.78·23-s + 3.98·25-s − 5.15·29-s + 4.66i·31-s + 2.99i·35-s − 6.10i·37-s − 12.3i·41-s − 4.45·43-s + 1.40·47-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 0.377i·7-s − 1.61i·11-s + 1.26i·13-s + 0.232i·17-s − 0.122·19-s + 0.581·23-s + 0.796·25-s − 0.957·29-s + 0.837i·31-s + 0.506i·35-s − 1.00i·37-s − 1.92i·41-s − 0.679·43-s + 0.204·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5100782741\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5100782741\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.99T + 5T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 - 4.55iT - 13T^{2} \) |
| 17 | \( 1 - 0.958iT - 17T^{2} \) |
| 19 | \( 1 + 0.532T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 + 5.15T + 29T^{2} \) |
| 31 | \( 1 - 4.66iT - 31T^{2} \) |
| 37 | \( 1 + 6.10iT - 37T^{2} \) |
| 41 | \( 1 + 12.3iT - 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 - 1.40T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 + 0.0356iT - 59T^{2} \) |
| 61 | \( 1 + 5.13iT - 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 + 1.90iT - 79T^{2} \) |
| 83 | \( 1 + 3.13iT - 83T^{2} \) |
| 89 | \( 1 - 5.91iT - 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.251348967197320247752606976128, −7.55856732620978947103442674773, −6.95173587225709701466019058560, −6.24735463339282905198832216383, −5.34604131592650494260093295220, −4.50847813665941966317267550141, −3.59986125122271389058536814154, −3.49320125451843986545112103084, −2.08066606567912451225111729454, −0.829137205884351128877993472752,
0.17345012531038593828825871023, 1.53793029119685332514583253212, 2.71166873379877481183959677701, 3.38760761296297749152305054613, 4.35048453081909068648248070972, 4.82363197748012797492920972734, 5.66214368879602515747603418355, 6.66276835419229007024957640452, 7.29635294998711200188334368492, 7.974072490260432784361105508677