| L(s) = 1 | + 3.62·5-s + 7-s − 3.91·11-s + 5.07·13-s − 1.03·17-s − 2.50·19-s − 4.95·23-s + 8.13·25-s + 9.20·29-s − 0.844·31-s + 3.62·35-s + 4.84·37-s + 4.14·41-s − 4.40·43-s + 7.87·47-s + 49-s − 12.2·53-s − 14.1·55-s + 11.2·59-s + 0.416·61-s + 18.3·65-s + 10.0·67-s + 5.05·71-s + 7.20·73-s − 3.91·77-s + 15.1·79-s − 1.86·83-s + ⋯ |
| L(s) = 1 | + 1.62·5-s + 0.377·7-s − 1.18·11-s + 1.40·13-s − 0.250·17-s − 0.575·19-s − 1.03·23-s + 1.62·25-s + 1.70·29-s − 0.151·31-s + 0.612·35-s + 0.796·37-s + 0.647·41-s − 0.671·43-s + 1.14·47-s + 0.142·49-s − 1.68·53-s − 1.91·55-s + 1.45·59-s + 0.0533·61-s + 2.28·65-s + 1.22·67-s + 0.599·71-s + 0.843·73-s − 0.446·77-s + 1.70·79-s − 0.204·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.858213008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.858213008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - 3.62T + 5T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 + 4.95T + 23T^{2} \) |
| 29 | \( 1 - 9.20T + 29T^{2} \) |
| 31 | \( 1 + 0.844T + 31T^{2} \) |
| 37 | \( 1 - 4.84T + 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 + 4.40T + 43T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 0.416T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 5.05T + 71T^{2} \) |
| 73 | \( 1 - 7.20T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 1.86T + 83T^{2} \) |
| 89 | \( 1 - 0.669T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.340352024214255916224930421222, −7.80641336908635938631750842618, −6.49011955369633952746386960086, −6.25966527950841600705803932683, −5.43558455913534634114763395750, −4.81540004578282422193976387279, −3.78029576294876543235646964754, −2.59569176031453020597128847403, −2.05952079497576312461526052922, −0.980046573994672445750962033329,
0.980046573994672445750962033329, 2.05952079497576312461526052922, 2.59569176031453020597128847403, 3.78029576294876543235646964754, 4.81540004578282422193976387279, 5.43558455913534634114763395750, 6.25966527950841600705803932683, 6.49011955369633952746386960086, 7.80641336908635938631750842618, 8.340352024214255916224930421222
