| L(s) = 1 | + 1.61·2-s + 0.928·3-s + 0.620·4-s − 0.0106·5-s + 1.50·6-s − 5.20·7-s − 2.23·8-s − 2.13·9-s − 0.0171·10-s + 0.534·11-s + 0.575·12-s − 4.21·13-s − 8.42·14-s − 0.00983·15-s − 4.85·16-s + 1.93·17-s − 3.46·18-s + 1.02·19-s − 0.00657·20-s − 4.83·21-s + 0.864·22-s + 1.90·23-s − 2.07·24-s − 4.99·25-s − 6.81·26-s − 4.76·27-s − 3.22·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.535·3-s + 0.310·4-s − 0.00474·5-s + 0.613·6-s − 1.96·7-s − 0.789·8-s − 0.712·9-s − 0.00542·10-s + 0.161·11-s + 0.166·12-s − 1.16·13-s − 2.25·14-s − 0.00254·15-s − 1.21·16-s + 0.469·17-s − 0.815·18-s + 0.234·19-s − 0.00147·20-s − 1.05·21-s + 0.184·22-s + 0.397·23-s − 0.423·24-s − 0.999·25-s − 1.33·26-s − 0.917·27-s − 0.610·28-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 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 0.928T + 3T^{2} \) |
| 5 | \( 1 + 0.0106T + 5T^{2} \) |
| 7 | \( 1 + 5.20T + 7T^{2} \) |
| 11 | \( 1 - 0.534T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 - 1.90T + 23T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 - 6.03T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 + 7.26T + 47T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 + 0.712T + 61T^{2} \) |
| 67 | \( 1 + 7.57T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 2.40T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
| 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
−9.641004118522715901999358333253, −8.921861570696351117388706767337, −7.83891731121247161858801891555, −6.66901395556845416875296392934, −6.11550625495215956860947571330, −5.17854919678193778115944338992, −4.06888158966560142313172659669, −3.06554366455891826924822209294, −2.75308484724707963253571896703, 0,
2.75308484724707963253571896703, 3.06554366455891826924822209294, 4.06888158966560142313172659669, 5.17854919678193778115944338992, 6.11550625495215956860947571330, 6.66901395556845416875296392934, 7.83891731121247161858801891555, 8.921861570696351117388706767337, 9.641004118522715901999358333253
