| L(s) = 1 | + 2.04·2-s + 3·3-s − 3.80·4-s + 16.1·5-s + 6.14·6-s − 1.13·7-s − 24.1·8-s + 9·9-s + 33.1·10-s + 36.3·11-s − 11.4·12-s + 78.1·13-s − 2.33·14-s + 48.5·15-s − 19.1·16-s − 43.6·17-s + 18.4·18-s − 18.4·19-s − 61.5·20-s − 3.41·21-s + 74.4·22-s − 4.45·23-s − 72.5·24-s + 136.·25-s + 160.·26-s + 27·27-s + 4.33·28-s + ⋯ |
| L(s) = 1 | + 0.724·2-s + 0.577·3-s − 0.475·4-s + 1.44·5-s + 0.418·6-s − 0.0615·7-s − 1.06·8-s + 0.333·9-s + 1.04·10-s + 0.995·11-s − 0.274·12-s + 1.66·13-s − 0.0445·14-s + 0.835·15-s − 0.299·16-s − 0.623·17-s + 0.241·18-s − 0.222·19-s − 0.687·20-s − 0.0355·21-s + 0.721·22-s − 0.0403·23-s − 0.617·24-s + 1.09·25-s + 1.20·26-s + 0.192·27-s + 0.0292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.274116022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.274116022\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 2.04T + 8T^{2} \) |
| 5 | \( 1 - 16.1T + 125T^{2} \) |
| 7 | \( 1 + 1.13T + 343T^{2} \) |
| 11 | \( 1 - 36.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 4.45T + 1.21e4T^{2} \) |
| 29 | \( 1 + 161.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 168.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 878.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 361.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 761.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 404.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 493.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 398.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−12.69361092024778055243033353567, −11.34430483539179120365567535044, −10.01509007829104527896177156308, −9.147327395913674794090075787854, −8.521572267683515180099923037091, −6.51766502691944559645801961406, −5.87161961057290297224819379821, −4.44347615582221479879858583209, −3.25581763744218795927894628597, −1.59192764700765547128510977036,
1.59192764700765547128510977036, 3.25581763744218795927894628597, 4.44347615582221479879858583209, 5.87161961057290297224819379821, 6.51766502691944559645801961406, 8.521572267683515180099923037091, 9.147327395913674794090075787854, 10.01509007829104527896177156308, 11.34430483539179120365567535044, 12.69361092024778055243033353567
