| L(s) = 1 | + (−1.59 − 1.57i)5-s − 2.43i·7-s + 0.884·11-s − 5.10i·13-s − 0.366i·17-s + 2.79·19-s + i·23-s + (0.0570 + 4.99i)25-s + 8.02·29-s + 7.24·31-s + (−3.82 + 3.86i)35-s − 3.10i·37-s + 3.47·41-s − 8.56i·43-s − 5.25i·47-s + ⋯ |
| L(s) = 1 | + (−0.711 − 0.703i)5-s − 0.919i·7-s + 0.266·11-s − 1.41i·13-s − 0.0889i·17-s + 0.642·19-s + 0.208i·23-s + (0.0114 + 0.999i)25-s + 1.49·29-s + 1.30·31-s + (−0.646 + 0.653i)35-s − 0.509i·37-s + 0.542·41-s − 1.30i·43-s − 0.766i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464109005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464109005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.59 + 1.57i)T \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 + 2.43iT - 7T^{2} \) |
| 11 | \( 1 - 0.884T + 11T^{2} \) |
| 13 | \( 1 + 5.10iT - 13T^{2} \) |
| 17 | \( 1 + 0.366iT - 17T^{2} \) |
| 19 | \( 1 - 2.79T + 19T^{2} \) |
| 29 | \( 1 - 8.02T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 3.10iT - 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 + 8.56iT - 43T^{2} \) |
| 47 | \( 1 + 5.25iT - 47T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 1.49iT - 67T^{2} \) |
| 71 | \( 1 + 8.29T + 71T^{2} \) |
| 73 | \( 1 - 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 6.06T + 79T^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 6.55iT - 97T^{2} \) |
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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
−8.312233303491579630472889972151, −7.39458925131027815700873672736, −6.93684704111550269342771849214, −5.79457084607303517819109625893, −5.10092952441535254721031441132, −4.31135624224453758534627844161, −3.61257302288692225719677870094, −2.76292280102771823253864774423, −1.18765287134076512776041203395, −0.49889585221283479696855196283,
1.29299168939455527259032758138, 2.57507791264583602535918907254, 3.08681917097964543476775959960, 4.31470921313180299092924341086, 4.68001992066997806332462131049, 6.04048712379210704821360929383, 6.40618637421551984549784466568, 7.21994391603948009779985880171, 7.966511452419269565075460593320, 8.680453966383786632929618492556
