L(s) = 1 | − 2.34·2-s + 5.73·3-s + 1.51·4-s − 13.4·6-s + 9.56i·7-s + 5.83·8-s + 23.9·9-s + 4.15·11-s + 8.68·12-s + 4.55·13-s − 22.4i·14-s − 19.7·16-s − 27.9i·17-s − 56.1·18-s + (−7.04 + 17.6i)19-s + ⋯ |
L(s) = 1 | − 1.17·2-s + 1.91·3-s + 0.378·4-s − 2.24·6-s + 1.36i·7-s + 0.729·8-s + 2.65·9-s + 0.377·11-s + 0.723·12-s + 0.350·13-s − 1.60i·14-s − 1.23·16-s − 1.64i·17-s − 3.11·18-s + (−0.370 + 0.928i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.886814825\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886814825\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (7.04 - 17.6i)T \) |
good | 2 | \( 1 + 2.34T + 4T^{2} \) |
| 3 | \( 1 - 5.73T + 9T^{2} \) |
| 7 | \( 1 - 9.56iT - 49T^{2} \) |
| 11 | \( 1 - 4.15T + 121T^{2} \) |
| 13 | \( 1 - 4.55T + 169T^{2} \) |
| 17 | \( 1 + 27.9iT - 289T^{2} \) |
| 23 | \( 1 - 13.9iT - 529T^{2} \) |
| 29 | \( 1 - 14.6iT - 841T^{2} \) |
| 31 | \( 1 - 28.9iT - 961T^{2} \) |
| 37 | \( 1 + 30.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 44.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 69.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 30.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 54.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 31.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 99.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.05T + 4.48e3T^{2} \) |
| 71 | \( 1 + 23.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 21.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 77.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 93.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 77.2T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34818413356758210684686972744, −9.572441407911735242717407117824, −8.936831105557496962215334377713, −8.579207392800050810766313508261, −7.69463551448674341648795097385, −6.84129219125442838782618032163, −5.07168536949928243044519383601, −3.66659543216351560446532200677, −2.53207918687467858759994304214, −1.53319696036885482134928275294,
1.05208614103589655091480784918, 2.20217946646891579911960097550, 3.82088857827399345437945964642, 4.28526371377030542807711857753, 6.74080896626589357082404775132, 7.41885444937890029344029865834, 8.335919330695138953565373828205, 8.663465915391762747627015216385, 9.705129980365436288757936534687, 10.25615239410701691812796322928