| L(s) = 1 | + 0.254·2-s − 3·3-s − 7.93·4-s − 10.8·5-s − 0.763·6-s − 23.2·7-s − 4.05·8-s + 9·9-s − 2.75·10-s + 51.0·11-s + 23.8·12-s + 51.8·13-s − 5.92·14-s + 32.4·15-s + 62.4·16-s − 0.0421·17-s + 2.28·18-s − 85.2·19-s + 85.8·20-s + 69.8·21-s + 12.9·22-s − 8.50·23-s + 12.1·24-s − 7.87·25-s + 13.1·26-s − 27·27-s + 184.·28-s + ⋯ |
| L(s) = 1 | + 0.0899·2-s − 0.577·3-s − 0.991·4-s − 0.967·5-s − 0.0519·6-s − 1.25·7-s − 0.179·8-s + 0.333·9-s − 0.0870·10-s + 1.39·11-s + 0.572·12-s + 1.10·13-s − 0.113·14-s + 0.558·15-s + 0.975·16-s − 0.000601·17-s + 0.0299·18-s − 1.02·19-s + 0.960·20-s + 0.726·21-s + 0.125·22-s − 0.0771·23-s + 0.103·24-s − 0.0629·25-s + 0.0995·26-s − 0.192·27-s + 1.24·28-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
| \(L(2)\) |
\(\approx\) |
\(0.7238131545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7238131545\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.254T + 8T^{2} \) |
| 5 | \( 1 + 10.8T + 125T^{2} \) |
| 7 | \( 1 + 23.2T + 343T^{2} \) |
| 11 | \( 1 - 51.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.0421T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 8.50T + 1.21e4T^{2} \) |
| 29 | \( 1 + 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 271.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.54T + 5.06e4T^{2} \) |
| 41 | \( 1 + 185.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 273.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 751.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 651.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 842.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 368.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 634.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 316.T + 9.12e5T^{2} \) |
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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
−12.27668686885522155742338087688, −11.41940617268964662964489853589, −10.19130674825441353010004172234, −9.186649301197940382953391180470, −8.313429374314832524075528244344, −6.78098478081726529182643603989, −5.92270743709261570084638303101, −4.21746566999097889094382420512, −3.65393940995306566276635447984, −0.68288059924350781120312986834,
0.68288059924350781120312986834, 3.65393940995306566276635447984, 4.21746566999097889094382420512, 5.92270743709261570084638303101, 6.78098478081726529182643603989, 8.313429374314832524075528244344, 9.186649301197940382953391180470, 10.19130674825441353010004172234, 11.41940617268964662964489853589, 12.27668686885522155742338087688
