| L(s) = 1 | − 3.49·2-s − 19.7·4-s + 59.5·5-s + 96.8·7-s + 180.·8-s − 208.·10-s + 731.·11-s − 631.·13-s − 338.·14-s + 0.920·16-s − 75.7·17-s + 2.35·19-s − 1.17e3·20-s − 2.55e3·22-s − 529·23-s + 426.·25-s + 2.20e3·26-s − 1.91e3·28-s + 2.42e3·29-s − 349.·31-s − 5.79e3·32-s + 264.·34-s + 5.77e3·35-s + 8.31e3·37-s − 8.22·38-s + 1.07e4·40-s − 4.19e3·41-s + ⋯ |
| L(s) = 1 | − 0.617·2-s − 0.618·4-s + 1.06·5-s + 0.746·7-s + 0.999·8-s − 0.658·10-s + 1.82·11-s − 1.03·13-s − 0.461·14-s + 0.000899·16-s − 0.0636·17-s + 0.00149·19-s − 0.659·20-s − 1.12·22-s − 0.208·23-s + 0.136·25-s + 0.640·26-s − 0.461·28-s + 0.535·29-s − 0.0653·31-s − 1.00·32-s + 0.0392·34-s + 0.796·35-s + 0.998·37-s − 0.000923·38-s + 1.06·40-s − 0.390·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.768050627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768050627\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 3.49T + 32T^{2} \) |
| 5 | \( 1 - 59.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 96.8T + 1.68e4T^{2} \) |
| 11 | \( 1 - 731.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 631.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 75.7T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.35T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.42e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 349.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.19e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.37e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 987.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
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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
−11.42788225306982401281143121779, −10.19439585579308163907718514878, −9.488436408600800148436575361041, −8.809047205588412369275925778449, −7.63946434256283661766111861894, −6.41181435333456567436290540954, −5.13700977436336546507746748249, −4.09341743042407156704099503769, −2.04170767010666991283045505902, −0.967008732056182976080419624387,
0.967008732056182976080419624387, 2.04170767010666991283045505902, 4.09341743042407156704099503769, 5.13700977436336546507746748249, 6.41181435333456567436290540954, 7.63946434256283661766111861894, 8.809047205588412369275925778449, 9.488436408600800148436575361041, 10.19439585579308163907718514878, 11.42788225306982401281143121779
