| L(s) = 1 | − 3-s + 5-s − 5.02·7-s + 9-s − 3.71·11-s − 6.49·13-s − 15-s + 5.37·17-s − 8.39·19-s + 5.02·21-s − 23-s + 25-s − 27-s + 3.31·29-s + 7.18·31-s + 3.71·33-s − 5.02·35-s − 3.02·37-s + 6.49·39-s + 0.782·41-s − 0.0950·43-s + 45-s + 8.39·47-s + 18.2·49-s − 5.37·51-s + 6.82·53-s − 3.71·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.89·7-s + 0.333·9-s − 1.11·11-s − 1.80·13-s − 0.258·15-s + 1.30·17-s − 1.92·19-s + 1.09·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.615·29-s + 1.28·31-s + 0.645·33-s − 0.848·35-s − 0.496·37-s + 1.03·39-s + 0.122·41-s − 0.0144·43-s + 0.149·45-s + 1.22·47-s + 2.60·49-s − 0.752·51-s + 0.937·53-s − 0.500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6186584847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6186584847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 5.02T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + 8.39T + 19T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 - 0.782T + 41T^{2} \) |
| 43 | \( 1 + 0.0950T + 43T^{2} \) |
| 47 | \( 1 - 8.39T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 9.71T + 67T^{2} \) |
| 71 | \( 1 - 8.73T + 71T^{2} \) |
| 73 | \( 1 - 3.64T + 73T^{2} \) |
| 79 | \( 1 - 8.59T + 79T^{2} \) |
| 83 | \( 1 + 7.37T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−8.989743081634208891406989594163, −7.942262779607835558233483775848, −7.12710303384092837626829920553, −6.47687281974739534478218001819, −5.81024413922984172315790400845, −5.07192283882270600359437462712, −4.11562134469833312760884684281, −2.90409052544495492707454519136, −2.37149458361629655917689449866, −0.46595732869095427329004366245,
0.46595732869095427329004366245, 2.37149458361629655917689449866, 2.90409052544495492707454519136, 4.11562134469833312760884684281, 5.07192283882270600359437462712, 5.81024413922984172315790400845, 6.47687281974739534478218001819, 7.12710303384092837626829920553, 7.942262779607835558233483775848, 8.989743081634208891406989594163
