| L(s) = 1 | + 0.912·2-s − 2.89·3-s − 1.16·4-s − 1.17·5-s − 2.64·6-s + 3.45·7-s − 2.89·8-s + 5.39·9-s − 1.07·10-s + 1.56·11-s + 3.37·12-s + 1.13·13-s + 3.15·14-s + 3.41·15-s − 0.305·16-s − 0.0116·17-s + 4.92·18-s + 2.31·19-s + 1.37·20-s − 10.0·21-s + 1.43·22-s − 7.71·23-s + 8.37·24-s − 3.61·25-s + 1.03·26-s − 6.92·27-s − 4.03·28-s + ⋯ |
| L(s) = 1 | + 0.645·2-s − 1.67·3-s − 0.583·4-s − 0.527·5-s − 1.07·6-s + 1.30·7-s − 1.02·8-s + 1.79·9-s − 0.340·10-s + 0.472·11-s + 0.975·12-s + 0.313·13-s + 0.843·14-s + 0.881·15-s − 0.0764·16-s − 0.00283·17-s + 1.16·18-s + 0.531·19-s + 0.307·20-s − 2.18·21-s + 0.304·22-s − 1.60·23-s + 1.70·24-s − 0.722·25-s + 0.202·26-s − 1.33·27-s − 0.762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
| good | 2 | \( 1 - 0.912T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 1.13T + 13T^{2} \) |
| 17 | \( 1 + 0.0116T + 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 - 2.15T + 29T^{2} \) |
| 31 | \( 1 + 8.15T + 31T^{2} \) |
| 37 | \( 1 + 8.91T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 + 4.59T + 43T^{2} \) |
| 47 | \( 1 - 3.42T + 47T^{2} \) |
| 53 | \( 1 - 2.09T + 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 + 5.07T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 3.42T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.806891463456142915927393649057, −8.665885658886780791753516008863, −7.81814362933705618221048153712, −6.78484372533395188204965692459, −5.72566474143078063636556443145, −5.30041517088737521706443854848, −4.38328700862298777615806811875, −3.77355762883754793672419023270, −1.54235238304647598460496460582, 0,
1.54235238304647598460496460582, 3.77355762883754793672419023270, 4.38328700862298777615806811875, 5.30041517088737521706443854848, 5.72566474143078063636556443145, 6.78484372533395188204965692459, 7.81814362933705618221048153712, 8.665885658886780791753516008863, 9.806891463456142915927393649057
