| L(s) = 1 | − 1.91·2-s + 1.82·3-s + 1.66·4-s + 2.84·5-s − 3.49·6-s − 3.81·7-s + 0.641·8-s + 0.330·9-s − 5.43·10-s − 2.90·11-s + 3.03·12-s + 0.331·13-s + 7.31·14-s + 5.18·15-s − 4.55·16-s − 6.31·17-s − 0.632·18-s + 0.291·19-s + 4.72·20-s − 6.97·21-s + 5.56·22-s − 7.19·23-s + 1.17·24-s + 3.06·25-s − 0.633·26-s − 4.87·27-s − 6.36·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 1.05·3-s + 0.832·4-s + 1.27·5-s − 1.42·6-s − 1.44·7-s + 0.226·8-s + 0.110·9-s − 1.71·10-s − 0.875·11-s + 0.877·12-s + 0.0918·13-s + 1.95·14-s + 1.33·15-s − 1.13·16-s − 1.53·17-s − 0.148·18-s + 0.0667·19-s + 1.05·20-s − 1.52·21-s + 1.18·22-s − 1.50·23-s + 0.238·24-s + 0.613·25-s − 0.124·26-s − 0.937·27-s − 1.20·28-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 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 0.331T + 13T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 19 | \( 1 - 0.291T + 19T^{2} \) |
| 23 | \( 1 + 7.19T + 23T^{2} \) |
| 29 | \( 1 + 5.72T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 + 4.31T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 3.94T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 0.154T + 89T^{2} \) |
| 97 | \( 1 - 3.35T + 97T^{2} \) |
| 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
−9.544252305732555873675738860728, −8.817343114885735874368601859631, −8.348996701608395918726332612129, −7.22429879166440666866012179804, −6.46056744351848150030048867871, −5.49880784169532122120632854685, −3.88964430410527708297190621787, −2.56424176226965894608361447633, −2.04144860628231338887918786217, 0,
2.04144860628231338887918786217, 2.56424176226965894608361447633, 3.88964430410527708297190621787, 5.49880784169532122120632854685, 6.46056744351848150030048867871, 7.22429879166440666866012179804, 8.348996701608395918726332612129, 8.817343114885735874368601859631, 9.544252305732555873675738860728
