| L(s) = 1 | + 1.34i·3-s + 5-s + (1.28 + 2.31i)7-s + 1.20·9-s − 2.44·11-s − 1.57·13-s + 1.34i·15-s + 1.11i·17-s + 8.44i·19-s + (−3.10 + 1.71i)21-s − 2.62i·23-s + 25-s + 5.63i·27-s − 3.43i·29-s + 9.70·31-s + ⋯ |
| L(s) = 1 | + 0.774i·3-s + 0.447·5-s + (0.483 + 0.875i)7-s + 0.400·9-s − 0.738·11-s − 0.435·13-s + 0.346i·15-s + 0.271i·17-s + 1.93i·19-s + (−0.677 + 0.374i)21-s − 0.548i·23-s + 0.200·25-s + 1.08i·27-s − 0.637i·29-s + 1.74·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.709215429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709215429\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.28 - 2.31i)T \) |
| good | 3 | \( 1 - 1.34iT - 3T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 1.11iT - 17T^{2} \) |
| 19 | \( 1 - 8.44iT - 19T^{2} \) |
| 23 | \( 1 + 2.62iT - 23T^{2} \) |
| 29 | \( 1 + 3.43iT - 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 + 6.22iT - 37T^{2} \) |
| 41 | \( 1 - 3.13iT - 41T^{2} \) |
| 43 | \( 1 + 7.45T + 43T^{2} \) |
| 47 | \( 1 + 9.40T + 47T^{2} \) |
| 53 | \( 1 - 11.6iT - 53T^{2} \) |
| 59 | \( 1 + 6.16iT - 59T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 + 7.87iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 7.70iT - 79T^{2} \) |
| 83 | \( 1 + 0.813iT - 83T^{2} \) |
| 89 | \( 1 - 5.12iT - 89T^{2} \) |
| 97 | \( 1 + 0.833iT - 97T^{2} \) |
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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
−10.02093593912759321194662638986, −9.510794199458839710340495109822, −8.354285620453132403287040405681, −7.909803860971922347211540537925, −6.56636266250512607898592076343, −5.66883357303190158637410535935, −4.95089593849193990050405055849, −4.06601577694126137840373048025, −2.80425640590324953434339056052, −1.71330490225660821064381925486,
0.76820735544013659653537704822, 1.97752294442255616107233471356, 3.08297682585180147480537498527, 4.61794594733182856135237829748, 5.10974161939629426908546727137, 6.59424338054672218360659366422, 7.00091372979984193328644009197, 7.82568580551800939237509791736, 8.595474427672402038775637769754, 9.814827593171973587686579156033
