| L(s) = 1 | − 2.31·2-s − 3-s + 3.37·4-s + 5-s + 2.31·6-s + 3.37·7-s − 3.19·8-s + 9-s − 2.31·10-s + 2.77·11-s − 3.37·12-s + 5.89·13-s − 7.83·14-s − 15-s + 0.646·16-s + 0.900·17-s − 2.31·18-s − 2.24·19-s + 3.37·20-s − 3.37·21-s − 6.44·22-s + 6.62·23-s + 3.19·24-s + 25-s − 13.6·26-s − 27-s + 11.4·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.577·3-s + 1.68·4-s + 0.447·5-s + 0.946·6-s + 1.27·7-s − 1.12·8-s + 0.333·9-s − 0.733·10-s + 0.837·11-s − 0.974·12-s + 1.63·13-s − 2.09·14-s − 0.258·15-s + 0.161·16-s + 0.218·17-s − 0.546·18-s − 0.516·19-s + 0.754·20-s − 0.737·21-s − 1.37·22-s + 1.38·23-s + 0.651·24-s + 0.200·25-s − 2.67·26-s − 0.192·27-s + 2.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.258952922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258952922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 0.900T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 6.62T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 + 4.44T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 - 6.80T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 + 7.19T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 - 6.96T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.251781148419929480103901768172, −7.56373545656766235368753017832, −6.68186763861894548488897308452, −6.33429980702236685775633027909, −5.34211724216485372434244573883, −4.57867870860394529816404316924, −3.55435065052579829153356424002, −2.21450691820566960328636710372, −1.35537500275979418705999254721, −0.940763796873746329152087135278,
0.940763796873746329152087135278, 1.35537500275979418705999254721, 2.21450691820566960328636710372, 3.55435065052579829153356424002, 4.57867870860394529816404316924, 5.34211724216485372434244573883, 6.33429980702236685775633027909, 6.68186763861894548488897308452, 7.56373545656766235368753017832, 8.251781148419929480103901768172
