| 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 | \( 1 + 77.2T + 9.40e3T^{2} \) |
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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
−10.25615239410701691812796322928, −9.705129980365436288757936534687, −8.663465915391762747627015216385, −8.335919330695138953565373828205, −7.41885444937890029344029865834, −6.74080896626589357082404775132, −4.28526371377030542807711857753, −3.82088857827399345437945964642, −2.20217946646891579911960097550, −1.05208614103589655091480784918,
1.53319696036885482134928275294, 2.53207918687467858759994304214, 3.66659543216351560446532200677, 5.07168536949928243044519383601, 6.84129219125442838782618032163, 7.69463551448674341648795097385, 8.579207392800050810766313508261, 8.936831105557496962215334377713, 9.572441407911735242717407117824, 10.34818413356758210684686972744
