| L(s) = 1 | + 3-s − 2·4-s + 5-s + 7-s + 9-s − 2·12-s + 4·13-s + 15-s + 4·16-s + 7·17-s + 5·19-s − 2·20-s + 21-s − 23-s + 25-s + 27-s − 2·28-s + 29-s + 4·31-s + 35-s − 2·36-s − 4·37-s + 4·39-s − 2·41-s + 4·43-s + 45-s + 2·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 1.10·13-s + 0.258·15-s + 16-s + 1.69·17-s + 1.14·19-s − 0.447·20-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 0.185·29-s + 0.718·31-s + 0.169·35-s − 1/3·36-s − 0.657·37-s + 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.060321389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.060321389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.64051610652568, −16.05008715830527, −15.51119472686265, −14.59284731749882, −14.27980653896715, −13.88237323507778, −13.29922714604270, −12.79668579791237, −12.03515469396265, −11.52849424999425, −10.51434554992851, −10.05442680859288, −9.567999659834626, −8.893294039913652, −8.313661181080024, −7.880172398374342, −7.135067759534869, −6.152908799924632, −5.461896824539060, −5.031564171157833, −4.007362939697951, −3.513093847457872, −2.756791909076876, −1.471877597352716, −0.9326028953344653,
0.9326028953344653, 1.471877597352716, 2.756791909076876, 3.513093847457872, 4.007362939697951, 5.031564171157833, 5.461896824539060, 6.152908799924632, 7.135067759534869, 7.880172398374342, 8.313661181080024, 8.893294039913652, 9.567999659834626, 10.05442680859288, 10.51434554992851, 11.52849424999425, 12.03515469396265, 12.79668579791237, 13.29922714604270, 13.88237323507778, 14.27980653896715, 14.59284731749882, 15.51119472686265, 16.05008715830527, 16.64051610652568
