| L(s) = 1 | − 0.156·2-s − 1.97·4-s + 3.65·5-s + 2.57·7-s + 0.623·8-s − 0.573·10-s − 2.56·11-s + 0.573·13-s − 0.403·14-s + 3.85·16-s − 0.780·17-s + 2·19-s − 7.22·20-s + 0.401·22-s + 2.56·23-s + 8.37·25-s − 0.0899·26-s − 5.08·28-s − 6.53·29-s + 6.80·31-s − 1.85·32-s + 0.122·34-s + 9.41·35-s + 6·37-s − 0.313·38-s + 2.27·40-s − 6.59·41-s + ⋯ |
| L(s) = 1 | − 0.110·2-s − 0.987·4-s + 1.63·5-s + 0.972·7-s + 0.220·8-s − 0.181·10-s − 0.773·11-s + 0.159·13-s − 0.107·14-s + 0.963·16-s − 0.189·17-s + 0.458·19-s − 1.61·20-s + 0.0857·22-s + 0.534·23-s + 1.67·25-s − 0.0176·26-s − 0.960·28-s − 1.21·29-s + 1.22·31-s − 0.327·32-s + 0.0209·34-s + 1.59·35-s + 0.986·37-s − 0.0508·38-s + 0.360·40-s − 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.587280450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587280450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.156T + 2T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 0.573T + 13T^{2} \) |
| 17 | \( 1 + 0.780T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 + 6.53T + 29T^{2} \) |
| 31 | \( 1 - 6.80T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 6.80T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 2.96T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 67 | \( 1 - 1.42T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 4.80T + 97T^{2} \) |
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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
−10.51981934793000241715632269250, −9.892465745070016011992149558389, −9.090107649758489566651661725705, −8.344284560457216544650089095388, −7.30898878541182454235537407715, −5.84724251882070557051676155604, −5.31131358727908220104930526428, −4.37196724091690988009834608521, −2.65293858688259224293994083927, −1.33121782324095828013283306741,
1.33121782324095828013283306741, 2.65293858688259224293994083927, 4.37196724091690988009834608521, 5.31131358727908220104930526428, 5.84724251882070557051676155604, 7.30898878541182454235537407715, 8.344284560457216544650089095388, 9.090107649758489566651661725705, 9.892465745070016011992149558389, 10.51981934793000241715632269250
