| L(s) = 1 | + 2.64·2-s − 0.999·4-s + 2.64·5-s − 23.8·8-s + 7.00·10-s − 18.5·11-s + 26·13-s − 55.0·16-s + 63.4·17-s + 35·19-s − 2.64·20-s − 49.0·22-s + 103.·23-s − 118·25-s + 68.7·26-s + 5.29·29-s + 75·31-s + 44.9·32-s + 168.·34-s − 111·37-s + 92.6·38-s − 63.0·40-s − 478.·41-s − 328·43-s + 18.5·44-s + 273·46-s − 380.·47-s + ⋯ |
| L(s) = 1 | + 0.935·2-s − 0.124·4-s + 0.236·5-s − 1.05·8-s + 0.221·10-s − 0.507·11-s + 0.554·13-s − 0.859·16-s + 0.905·17-s + 0.422·19-s − 0.0295·20-s − 0.474·22-s + 0.935·23-s − 0.944·25-s + 0.518·26-s + 0.0338·29-s + 0.434·31-s + 0.248·32-s + 0.847·34-s − 0.493·37-s + 0.395·38-s − 0.249·40-s − 1.82·41-s − 1.16·43-s + 0.0634·44-s + 0.875·46-s − 1.18·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 8T^{2} \) |
| 5 | \( 1 - 2.64T + 125T^{2} \) |
| 11 | \( 1 + 18.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.29T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75T + 2.97e4T^{2} \) |
| 37 | \( 1 + 111T + 5.06e4T^{2} \) |
| 41 | \( 1 + 478.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 328T + 7.95e4T^{2} \) |
| 47 | \( 1 + 380.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 126.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 126.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 152T + 2.26e5T^{2} \) |
| 67 | \( 1 - 202T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 672T + 3.89e5T^{2} \) |
| 79 | \( 1 + 988T + 4.93e5T^{2} \) |
| 83 | \( 1 - 142.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 965.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 492T + 9.12e5T^{2} \) |
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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
−8.797268946056108669295992221313, −8.117603682197776055954077486835, −7.03265366961736394723888699911, −6.09698458680685708034357552065, −5.36703454935999894909795153887, −4.71687391554463222607849750224, −3.56295450429409512934943277747, −2.96616582901634837484482309997, −1.49000393028158028774334907925, 0,
1.49000393028158028774334907925, 2.96616582901634837484482309997, 3.56295450429409512934943277747, 4.71687391554463222607849750224, 5.36703454935999894909795153887, 6.09698458680685708034357552065, 7.03265366961736394723888699911, 8.117603682197776055954077486835, 8.797268946056108669295992221313
