| L(s) = 1 | + 0.290·2-s + 3-s − 1.91·4-s + 5-s + 0.290·6-s − 0.486·7-s − 1.13·8-s + 9-s + 0.290·10-s + 5.34·11-s − 1.91·12-s + 2.48·13-s − 0.141·14-s + 15-s + 3.50·16-s − 3.41·17-s + 0.290·18-s − 1.91·20-s − 0.486·21-s + 1.55·22-s + 6.41·23-s − 1.13·24-s + 25-s + 0.722·26-s + 27-s + 0.932·28-s − 2.76·29-s + ⋯ |
| L(s) = 1 | + 0.205·2-s + 0.577·3-s − 0.957·4-s + 0.447·5-s + 0.118·6-s − 0.183·7-s − 0.402·8-s + 0.333·9-s + 0.0919·10-s + 1.61·11-s − 0.552·12-s + 0.689·13-s − 0.0378·14-s + 0.258·15-s + 0.875·16-s − 0.828·17-s + 0.0685·18-s − 0.428·20-s − 0.106·21-s + 0.331·22-s + 1.33·23-s − 0.232·24-s + 0.200·25-s + 0.141·26-s + 0.192·27-s + 0.176·28-s − 0.513·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.657276388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657276388\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.290T + 2T^{2} \) |
| 7 | \( 1 + 0.486T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 23 | \( 1 - 6.41T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 + 4.83T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 2.99T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 8.13T + 79T^{2} \) |
| 83 | \( 1 + 5.86T + 83T^{2} \) |
| 89 | \( 1 - 6.51T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.528787460004892007838794396527, −7.43954882266416155270524430341, −6.66312812110099162460583336118, −6.06521841688150259079659945990, −5.18343317752888293396534787999, −4.33571019733651494623673640432, −3.77030742875690921237359535823, −3.04625630065028179613573610342, −1.81964552155927779464209960618, −0.882199977243032778929710025783,
0.882199977243032778929710025783, 1.81964552155927779464209960618, 3.04625630065028179613573610342, 3.77030742875690921237359535823, 4.33571019733651494623673640432, 5.18343317752888293396534787999, 6.06521841688150259079659945990, 6.66312812110099162460583336118, 7.43954882266416155270524430341, 8.528787460004892007838794396527
