L(s) = 1 | − 0.254·2-s − 3-s − 1.93·4-s − 1.93·5-s + 0.254·6-s − 7-s + 8-s + 9-s + 0.491·10-s + 3.87·11-s + 1.93·12-s + 5.68·13-s + 0.254·14-s + 1.93·15-s + 3.61·16-s − 5.69·17-s − 0.254·18-s + 2.31·19-s + 3.74·20-s + 21-s − 0.983·22-s − 3.50·23-s − 24-s − 1.25·25-s − 1.44·26-s − 27-s + 1.93·28-s + ⋯ |
L(s) = 1 | − 0.179·2-s − 0.577·3-s − 0.967·4-s − 0.865·5-s + 0.103·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.155·10-s + 1.16·11-s + 0.558·12-s + 1.57·13-s + 0.0679·14-s + 0.499·15-s + 0.904·16-s − 1.38·17-s − 0.0598·18-s + 0.531·19-s + 0.837·20-s + 0.218·21-s − 0.209·22-s − 0.731·23-s − 0.204·24-s − 0.250·25-s − 0.283·26-s − 0.192·27-s + 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7480942022\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7480942022\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 + 0.254T + 2T^{2} \) |
| 5 | \( 1 + 1.93T + 5T^{2} \) |
| 11 | \( 1 - 3.87T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 + 3.50T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 0.826T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.38T + 61T^{2} \) |
| 67 | \( 1 + 3.06T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 + 9.69T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.457166762869705212916886207855, −7.904057512422382858565046125059, −6.85785141330466255198531989748, −6.29717176706829382061657800519, −5.51337100763001135609865203259, −4.38505883248875814239641039751, −4.04899434164127530734691093792, −3.33567700651976668361911816105, −1.59274361085997758472205017177, −0.55881691903650565761123273360,
0.55881691903650565761123273360, 1.59274361085997758472205017177, 3.33567700651976668361911816105, 4.04899434164127530734691093792, 4.38505883248875814239641039751, 5.51337100763001135609865203259, 6.29717176706829382061657800519, 6.85785141330466255198531989748, 7.904057512422382858565046125059, 8.457166762869705212916886207855