L(s) = 1 | − 2.42·2-s + 0.803·3-s + 3.86·4-s + 3.28·5-s − 1.94·6-s + 3.29·7-s − 4.51·8-s − 2.35·9-s − 7.96·10-s + 6.14·11-s + 3.10·12-s + 1.33·13-s − 7.98·14-s + 2.64·15-s + 3.19·16-s − 1.21·17-s + 5.69·18-s + 2.94·19-s + 12.7·20-s + 2.65·21-s − 14.8·22-s + 6.73·23-s − 3.62·24-s + 5.81·25-s − 3.24·26-s − 4.30·27-s + 12.7·28-s + ⋯ |
L(s) = 1 | − 1.71·2-s + 0.464·3-s + 1.93·4-s + 1.47·5-s − 0.794·6-s + 1.24·7-s − 1.59·8-s − 0.784·9-s − 2.51·10-s + 1.85·11-s + 0.896·12-s + 0.371·13-s − 2.13·14-s + 0.682·15-s + 0.798·16-s − 0.293·17-s + 1.34·18-s + 0.676·19-s + 2.84·20-s + 0.578·21-s − 3.17·22-s + 1.40·23-s − 0.739·24-s + 1.16·25-s − 0.636·26-s − 0.828·27-s + 2.40·28-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
\(L(1)\) |
\(\approx\) |
\(1.352084098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352084098\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 - 0.803T + 3T^{2} \) |
| 5 | \( 1 - 3.28T + 5T^{2} \) |
| 7 | \( 1 - 3.29T + 7T^{2} \) |
| 11 | \( 1 - 6.14T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 - 8.05T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 + 9.40T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 0.808T + 73T^{2} \) |
| 79 | \( 1 - 0.804T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 - 8.89T + 97T^{2} \) |
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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
−9.684030771978441819880530549313, −8.927246429007907094658837142961, −8.842385372859131285554686064699, −7.78209051911902577748986537503, −6.79392639292230214136049507714, −6.06574214002997802367611489221, −4.93635704791776916239574207198, −3.18120429393216270285703669655, −1.82336982030220048775893805455, −1.39800864815033171308811325076,
1.39800864815033171308811325076, 1.82336982030220048775893805455, 3.18120429393216270285703669655, 4.93635704791776916239574207198, 6.06574214002997802367611489221, 6.79392639292230214136049507714, 7.78209051911902577748986537503, 8.842385372859131285554686064699, 8.927246429007907094658837142961, 9.684030771978441819880530549313