L(s) = 1 | + 2.59·2-s − 2.74·3-s + 4.71·4-s + 3.63·5-s − 7.11·6-s + 3.18·7-s + 7.04·8-s + 4.54·9-s + 9.40·10-s − 2.94·11-s − 12.9·12-s − 1.90·13-s + 8.25·14-s − 9.96·15-s + 8.82·16-s − 17-s + 11.7·18-s + 3.08·19-s + 17.1·20-s − 8.74·21-s − 7.63·22-s − 7.87·23-s − 19.3·24-s + 8.17·25-s − 4.94·26-s − 4.23·27-s + 15.0·28-s + ⋯ |
L(s) = 1 | + 1.83·2-s − 1.58·3-s + 2.35·4-s + 1.62·5-s − 2.90·6-s + 1.20·7-s + 2.49·8-s + 1.51·9-s + 2.97·10-s − 0.888·11-s − 3.74·12-s − 0.528·13-s + 2.20·14-s − 2.57·15-s + 2.20·16-s − 0.242·17-s + 2.77·18-s + 0.707·19-s + 3.82·20-s − 1.90·21-s − 1.62·22-s − 1.64·23-s − 3.94·24-s + 1.63·25-s − 0.968·26-s − 0.814·27-s + 2.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.075109465\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.075109465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 + 1.90T + 13T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 + 7.87T + 23T^{2} \) |
| 29 | \( 1 + 0.623T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 1.13T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 - 5.03T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 61 | \( 1 + 8.78T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−10.66013962210804094883114459904, −9.575908839916431661898584919227, −7.81118126584702312872089845043, −6.89676202856989811593882659526, −5.97717279617701437061383151758, −5.49941665401812683891895726733, −5.10079431431854520484441431035, −4.24329450510981642616963628101, −2.48998298991182567311074057867, −1.64806429396503185880595473737,
1.64806429396503185880595473737, 2.48998298991182567311074057867, 4.24329450510981642616963628101, 5.10079431431854520484441431035, 5.49941665401812683891895726733, 5.97717279617701437061383151758, 6.89676202856989811593882659526, 7.81118126584702312872089845043, 9.575908839916431661898584919227, 10.66013962210804094883114459904