| L(s) = 1 | + 1.23·2-s + 81·3-s − 510.·4-s − 1.72e3·5-s + 99.9·6-s + 4.57e3·7-s − 1.26e3·8-s + 6.56e3·9-s − 2.12e3·10-s − 7.56e4·11-s − 4.13e4·12-s − 1.28e5·13-s + 5.65e3·14-s − 1.39e5·15-s + 2.59e5·16-s − 7.94e3·17-s + 8.09e3·18-s − 2.91e5·19-s + 8.79e5·20-s + 3.70e5·21-s − 9.33e4·22-s − 5.86e5·23-s − 1.02e5·24-s + 1.01e6·25-s − 1.59e5·26-s + 5.31e5·27-s − 2.33e6·28-s + ⋯ |
| L(s) = 1 | + 0.0545·2-s + 0.577·3-s − 0.997·4-s − 1.23·5-s + 0.0314·6-s + 0.720·7-s − 0.108·8-s + 0.333·9-s − 0.0672·10-s − 1.55·11-s − 0.575·12-s − 1.25·13-s + 0.0393·14-s − 0.711·15-s + 0.991·16-s − 0.0230·17-s + 0.0181·18-s − 0.513·19-s + 1.22·20-s + 0.416·21-s − 0.0849·22-s − 0.436·23-s − 0.0628·24-s + 0.518·25-s − 0.0682·26-s + 0.192·27-s − 0.718·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6100569661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6100569661\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 1.23T + 512T^{2} \) |
| 5 | \( 1 + 1.72e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.57e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.56e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.28e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 7.94e3T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.91e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.86e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.07e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.13e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.10e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.55e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.62e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 5.77e5T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.20e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.52e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.25e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.68e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.60e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.26e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.81e8T + 7.60e17T^{2} \) |
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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
−10.95074633690262211124091745696, −9.959771851936681497257936318115, −8.793210419269309315579870814161, −7.88657814681465688660742343037, −7.49835338530664893668254435566, −5.32449178263742103857248685580, −4.53722901958951024230134487544, −3.54358410357225273106957077132, −2.20109354769548604789409875274, −0.35603105079820288194391628244,
0.35603105079820288194391628244, 2.20109354769548604789409875274, 3.54358410357225273106957077132, 4.53722901958951024230134487544, 5.32449178263742103857248685580, 7.49835338530664893668254435566, 7.88657814681465688660742343037, 8.793210419269309315579870814161, 9.959771851936681497257936318115, 10.95074633690262211124091745696
