| L(s) = 1 | + 0.607·2-s − 3·3-s − 7.63·4-s − 1.82·6-s − 8.95·7-s − 9.49·8-s + 9·9-s − 11·11-s + 22.8·12-s + 0.460·13-s − 5.43·14-s + 55.2·16-s − 128.·17-s + 5.46·18-s − 0.0245·19-s + 26.8·21-s − 6.67·22-s − 171.·23-s + 28.4·24-s + 0.279·26-s − 27·27-s + 68.3·28-s − 226.·29-s + 195.·31-s + 109.·32-s + 33·33-s − 77.9·34-s + ⋯ |
| L(s) = 1 | + 0.214·2-s − 0.577·3-s − 0.953·4-s − 0.123·6-s − 0.483·7-s − 0.419·8-s + 0.333·9-s − 0.301·11-s + 0.550·12-s + 0.00982·13-s − 0.103·14-s + 0.863·16-s − 1.83·17-s + 0.0715·18-s − 0.000296·19-s + 0.279·21-s − 0.0647·22-s − 1.55·23-s + 0.242·24-s + 0.00210·26-s − 0.192·27-s + 0.461·28-s − 1.45·29-s + 1.13·31-s + 0.604·32-s + 0.174·33-s − 0.393·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5687851482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5687851482\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 0.607T + 8T^{2} \) |
| 7 | \( 1 + 8.95T + 343T^{2} \) |
| 13 | \( 1 - 0.460T + 2.19e3T^{2} \) |
| 17 | \( 1 + 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.0245T + 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 622.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 9.86T + 2.05e5T^{2} \) |
| 61 | \( 1 - 902.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 146.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 893.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 459.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 125.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 150.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−9.900807162999756533711936002120, −9.009789794740617683620246881184, −8.293059132730993382224606221463, −7.12019393208585419538659128040, −6.19297851827383244175495823273, −5.39129010626140626323981043959, −4.42384484925535726599683533098, −3.67608447760003216773888503742, −2.15414247791255037928271882696, −0.40288930299521032155519409383,
0.40288930299521032155519409383, 2.15414247791255037928271882696, 3.67608447760003216773888503742, 4.42384484925535726599683533098, 5.39129010626140626323981043959, 6.19297851827383244175495823273, 7.12019393208585419538659128040, 8.293059132730993382224606221463, 9.009789794740617683620246881184, 9.900807162999756533711936002120
