| L(s) = 1 | + 1.12·2-s + 2.02·3-s − 0.732·4-s − 3.34·5-s + 2.27·6-s + 0.00528·7-s − 3.07·8-s + 1.08·9-s − 3.76·10-s + 1.36·11-s − 1.48·12-s − 2.90·13-s + 0.00595·14-s − 6.75·15-s − 1.99·16-s − 5.62·17-s + 1.22·18-s − 5.10·19-s + 2.44·20-s + 0.0106·21-s + 1.53·22-s + 0.143·23-s − 6.21·24-s + 6.16·25-s − 3.26·26-s − 3.87·27-s − 0.00387·28-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 1.16·3-s − 0.366·4-s − 1.49·5-s + 0.928·6-s + 0.00199·7-s − 1.08·8-s + 0.361·9-s − 1.18·10-s + 0.410·11-s − 0.427·12-s − 0.804·13-s + 0.00159·14-s − 1.74·15-s − 0.499·16-s − 1.36·17-s + 0.287·18-s − 1.17·19-s + 0.547·20-s + 0.00233·21-s + 0.326·22-s + 0.0299·23-s − 1.26·24-s + 1.23·25-s − 0.640·26-s − 0.744·27-s − 0.000732·28-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 | \( 1 - 1.12T + 2T^{2} \) |
| 3 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 - 0.00528T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 + 5.10T + 19T^{2} \) |
| 23 | \( 1 - 0.143T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 + 9.51T + 37T^{2} \) |
| 41 | \( 1 - 6.91T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 5.99T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 + 9.68T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 8.61T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 2.65T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−9.139383169423905178090803927024, −8.786240865425189271866950450896, −8.029589842418431565489190996578, −7.18086697630280526589559993801, −6.14497579904296210761842539560, −4.60048867044085602018335453236, −4.26874513008043591310978174847, −3.34185055551209971507124731758, −2.45279205971243211625320334304, 0,
2.45279205971243211625320334304, 3.34185055551209971507124731758, 4.26874513008043591310978174847, 4.60048867044085602018335453236, 6.14497579904296210761842539560, 7.18086697630280526589559993801, 8.029589842418431565489190996578, 8.786240865425189271866950450896, 9.139383169423905178090803927024
