| L(s) = 1 | + 1.45·2-s − 0.0876·3-s + 0.102·4-s + 0.663·5-s − 0.127·6-s − 2.02·7-s − 2.75·8-s − 2.99·9-s + 0.962·10-s − 2.07·11-s − 0.00899·12-s + 5.02·13-s − 2.92·14-s − 0.0581·15-s − 4.19·16-s − 3.40·17-s − 4.33·18-s − 4.98·19-s + 0.0680·20-s + 0.177·21-s − 3.00·22-s + 2.01·23-s + 0.241·24-s − 4.55·25-s + 7.28·26-s + 0.525·27-s − 0.207·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s − 0.0506·3-s + 0.0512·4-s + 0.296·5-s − 0.0518·6-s − 0.763·7-s − 0.972·8-s − 0.997·9-s + 0.304·10-s − 0.625·11-s − 0.00259·12-s + 1.39·13-s − 0.782·14-s − 0.0150·15-s − 1.04·16-s − 0.824·17-s − 1.02·18-s − 1.14·19-s + 0.0152·20-s + 0.0386·21-s − 0.641·22-s + 0.419·23-s + 0.0492·24-s − 0.911·25-s + 1.42·26-s + 0.101·27-s − 0.0391·28-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 | \( 1 - 1.45T + 2T^{2} \) |
| 3 | \( 1 + 0.0876T + 3T^{2} \) |
| 5 | \( 1 - 0.663T + 5T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 2.01T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 + 7.18T + 53T^{2} \) |
| 59 | \( 1 + 3.57T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 6.81T + 67T^{2} \) |
| 71 | \( 1 - 9.64T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 + 7.78T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 0.490T + 97T^{2} \) |
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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
−9.343907958506482525920624974025, −8.927902292497007901050093534217, −7.976061188988506192235102665642, −6.51171158658257538879959132366, −6.04036935303615748490762938315, −5.31334356183668398723710475808, −4.14235126016244013573277524405, −3.35234710386843700961891549410, −2.32599681120643725523791439826, 0,
2.32599681120643725523791439826, 3.35234710386843700961891549410, 4.14235126016244013573277524405, 5.31334356183668398723710475808, 6.04036935303615748490762938315, 6.51171158658257538879959132366, 7.976061188988506192235102665642, 8.927902292497007901050093534217, 9.343907958506482525920624974025
