L(s) = 1 | − 2.55·2-s + 2.02·3-s + 4.52·4-s − 4.30·5-s − 5.16·6-s + 1.54·7-s − 6.43·8-s + 1.08·9-s + 10.9·10-s + 5.86·11-s + 9.14·12-s + 3.29·13-s − 3.94·14-s − 8.70·15-s + 7.39·16-s − 2.14·17-s − 2.78·18-s − 1.66·19-s − 19.4·20-s + 3.12·21-s − 14.9·22-s − 1.83·23-s − 13.0·24-s + 13.5·25-s − 8.40·26-s − 3.86·27-s + 6.98·28-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 1.16·3-s + 2.26·4-s − 1.92·5-s − 2.10·6-s + 0.583·7-s − 2.27·8-s + 0.363·9-s + 3.47·10-s + 1.76·11-s + 2.63·12-s + 0.912·13-s − 1.05·14-s − 2.24·15-s + 1.84·16-s − 0.519·17-s − 0.655·18-s − 0.382·19-s − 4.35·20-s + 0.681·21-s − 3.19·22-s − 0.381·23-s − 2.65·24-s + 2.70·25-s − 1.64·26-s − 0.743·27-s + 1.31·28-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
\(L(1)\) |
\(\approx\) |
\(0.8186515291\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8186515291\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 13 | \( 1 - 3.29T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 - 8.97T + 41T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 - 3.44T + 47T^{2} \) |
| 53 | \( 1 - 5.89T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 0.264T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 - 8.77T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 - 7.54T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−9.529602393021805453876851476011, −8.819560162209189548401707791106, −8.566209047680225946701796544967, −7.75031309756824667247461414395, −7.29198667292911285429509459947, −6.28994441791368420810120872834, −4.10785982019730710283269901542, −3.61384922598407807576691399372, −2.19416100434147895973647190019, −0.896382273047701522959466557081,
0.896382273047701522959466557081, 2.19416100434147895973647190019, 3.61384922598407807576691399372, 4.10785982019730710283269901542, 6.28994441791368420810120872834, 7.29198667292911285429509459947, 7.75031309756824667247461414395, 8.566209047680225946701796544967, 8.819560162209189548401707791106, 9.529602393021805453876851476011