| L(s) = 1 | − 2-s − 3.38·3-s + 4-s − 3.95·5-s + 3.38·6-s − 2.93·7-s − 8-s + 8.48·9-s + 3.95·10-s − 5.00·11-s − 3.38·12-s + 4.79·13-s + 2.93·14-s + 13.3·15-s + 16-s − 1.70·17-s − 8.48·18-s − 0.841·19-s − 3.95·20-s + 9.95·21-s + 5.00·22-s − 0.697·23-s + 3.38·24-s + 10.6·25-s − 4.79·26-s − 18.5·27-s − 2.93·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.95·3-s + 0.5·4-s − 1.76·5-s + 1.38·6-s − 1.11·7-s − 0.353·8-s + 2.82·9-s + 1.24·10-s − 1.50·11-s − 0.978·12-s + 1.33·13-s + 0.785·14-s + 3.45·15-s + 0.250·16-s − 0.413·17-s − 1.99·18-s − 0.193·19-s − 0.883·20-s + 2.17·21-s + 1.06·22-s − 0.145·23-s + 0.691·24-s + 2.12·25-s − 0.940·26-s − 3.57·27-s − 0.555·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05431240111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05431240111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 3.38T + 3T^{2} \) |
| 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 5.00T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 0.841T + 19T^{2} \) |
| 23 | \( 1 + 0.697T + 23T^{2} \) |
| 29 | \( 1 - 2.42T + 29T^{2} \) |
| 37 | \( 1 + 0.742T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 + 7.50T + 53T^{2} \) |
| 59 | \( 1 - 5.41T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.64T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 6.28T + 73T^{2} \) |
| 79 | \( 1 + 4.50T + 79T^{2} \) |
| 83 | \( 1 + 7.82T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
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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
−7.78402562223794498765780087763, −7.45590783434867674531675840351, −6.56491430677252843433965386023, −6.18709437439243501804580622786, −5.34254227523161791325298478575, −4.45771126597294154529319710195, −3.82590241485957917978743608826, −2.86156377407650774959733530768, −1.18004715165501106042649785564, −0.17537798921557620245806996226,
0.17537798921557620245806996226, 1.18004715165501106042649785564, 2.86156377407650774959733530768, 3.82590241485957917978743608826, 4.45771126597294154529319710195, 5.34254227523161791325298478575, 6.18709437439243501804580622786, 6.56491430677252843433965386023, 7.45590783434867674531675840351, 7.78402562223794498765780087763
