| L(s) = 1 | − 2-s + 0.592·3-s + 4-s − 0.448·5-s − 0.592·6-s + 3.30·7-s − 8-s − 2.64·9-s + 0.448·10-s + 0.273·11-s + 0.592·12-s − 1.38·13-s − 3.30·14-s − 0.265·15-s + 16-s + 2.43·17-s + 2.64·18-s + 19-s − 0.448·20-s + 1.95·21-s − 0.273·22-s − 3.01·23-s − 0.592·24-s − 4.79·25-s + 1.38·26-s − 3.34·27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.342·3-s + 0.5·4-s − 0.200·5-s − 0.241·6-s + 1.24·7-s − 0.353·8-s − 0.882·9-s + 0.141·10-s + 0.0823·11-s + 0.171·12-s − 0.383·13-s − 0.883·14-s − 0.0685·15-s + 0.250·16-s + 0.591·17-s + 0.624·18-s + 0.229·19-s − 0.100·20-s + 0.427·21-s − 0.0582·22-s − 0.627·23-s − 0.120·24-s − 0.959·25-s + 0.270·26-s − 0.644·27-s + 0.624·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.592T + 3T^{2} \) |
| 5 | \( 1 + 0.448T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 0.273T + 11T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 8.03T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 2.50T + 43T^{2} \) |
| 47 | \( 1 + 8.67T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 - 1.28T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 + 8.67T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + 5.87T + 97T^{2} \) |
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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
−7.77224900665188944157003861820, −7.09493302276665736773626732297, −6.04714651010441448711189985051, −5.52468762989789645763958112287, −4.68253060997261823447049848446, −3.82027524945148003735395670949, −2.91035909376718764216860136937, −2.12052249156508338368778243730, −1.30066089940505451408509431953, 0,
1.30066089940505451408509431953, 2.12052249156508338368778243730, 2.91035909376718764216860136937, 3.82027524945148003735395670949, 4.68253060997261823447049848446, 5.52468762989789645763958112287, 6.04714651010441448711189985051, 7.09493302276665736773626732297, 7.77224900665188944157003861820
