| L(s) = 1 | + 2.87·5-s − 7-s − 1.18·11-s − 4.75·13-s − 5.41·17-s − 1.10·19-s + 5.90·23-s + 3.29·25-s − 4.98·29-s + 5.57·31-s − 2.87·35-s − 2.42·37-s − 7.63·41-s + 7.57·43-s + 0.283·47-s + 49-s + 4.22·53-s − 3.41·55-s + 11.7·59-s + 1.98·61-s − 13.7·65-s + 13.5·67-s − 5.11·71-s − 0.327·73-s + 1.18·77-s + 10.4·79-s + 7.25·83-s + ⋯ |
| L(s) = 1 | + 1.28·5-s − 0.377·7-s − 0.357·11-s − 1.31·13-s − 1.31·17-s − 0.253·19-s + 1.23·23-s + 0.658·25-s − 0.925·29-s + 1.00·31-s − 0.486·35-s − 0.398·37-s − 1.19·41-s + 1.15·43-s + 0.0412·47-s + 0.142·49-s + 0.580·53-s − 0.460·55-s + 1.52·59-s + 0.254·61-s − 1.69·65-s + 1.65·67-s − 0.606·71-s − 0.0383·73-s + 0.135·77-s + 1.18·79-s + 0.796·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.005279846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005279846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.87T + 5T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 + 4.98T + 29T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 + 2.42T + 37T^{2} \) |
| 41 | \( 1 + 7.63T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 - 0.283T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 + 0.327T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.25T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.61853898542556140028033956548, −6.84265023592385566260119570534, −6.52420132860505270256062507958, −5.56467107358535976174956455477, −5.10808174913628658183882527390, −4.38489505943158945544014373528, −3.30281685730568073087897900673, −2.34069707002746933547932828639, −2.09257618724218220383161314607, −0.65187558950093180119691289059,
0.65187558950093180119691289059, 2.09257618724218220383161314607, 2.34069707002746933547932828639, 3.30281685730568073087897900673, 4.38489505943158945544014373528, 5.10808174913628658183882527390, 5.56467107358535976174956455477, 6.52420132860505270256062507958, 6.84265023592385566260119570534, 7.61853898542556140028033956548
