| L(s) = 1 | + 0.940·3-s + 2.53·5-s + 0.182·7-s − 2.11·9-s + 0.888·11-s + 0.134·13-s + 2.38·15-s + 5.15·17-s − 2.24·19-s + 0.171·21-s + 4.77·23-s + 1.44·25-s − 4.81·27-s − 3.24·29-s + 10.4·31-s + 0.835·33-s + 0.462·35-s + 6.14·37-s + 0.126·39-s + 0.268·41-s − 3.38·43-s − 5.37·45-s + 1.64·47-s − 6.96·49-s + 4.84·51-s − 0.125·53-s + 2.25·55-s + ⋯ |
| L(s) = 1 | + 0.542·3-s + 1.13·5-s + 0.0688·7-s − 0.705·9-s + 0.268·11-s + 0.0372·13-s + 0.616·15-s + 1.25·17-s − 0.515·19-s + 0.0373·21-s + 0.994·23-s + 0.289·25-s − 0.925·27-s − 0.602·29-s + 1.88·31-s + 0.145·33-s + 0.0782·35-s + 1.01·37-s + 0.0202·39-s + 0.0419·41-s − 0.516·43-s − 0.800·45-s + 0.239·47-s − 0.995·49-s + 0.678·51-s − 0.0172·53-s + 0.304·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.994316309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994316309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.940T + 3T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 - 0.182T + 7T^{2} \) |
| 11 | \( 1 - 0.888T + 11T^{2} \) |
| 13 | \( 1 - 0.134T + 13T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 6.14T + 37T^{2} \) |
| 41 | \( 1 - 0.268T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 1.64T + 47T^{2} \) |
| 53 | \( 1 + 0.125T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 - 1.17T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 5.14T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−8.344737678114483997946651713423, −7.985227418547051605215123415321, −6.88619058948072935915002952445, −6.16802463084239988646763191185, −5.56119717099198192275667484518, −4.79780894007240467420763070274, −3.65175870595627373205452301568, −2.84741004863274582218604362083, −2.10660247268211474366592085572, −1.00547151245030427243328596969,
1.00547151245030427243328596969, 2.10660247268211474366592085572, 2.84741004863274582218604362083, 3.65175870595627373205452301568, 4.79780894007240467420763070274, 5.56119717099198192275667484518, 6.16802463084239988646763191185, 6.88619058948072935915002952445, 7.985227418547051605215123415321, 8.344737678114483997946651713423
