| L(s) = 1 | + (−0.438 − 1.95i)2-s − 1.73i·3-s + (−3.61 + 1.71i)4-s + (−3.37 + 0.758i)6-s − 6.33i·7-s + (4.92 + 6.30i)8-s − 2.99·9-s − 9.27i·11-s + (2.96 + 6.26i)12-s − 18.5·13-s + (−12.3 + 2.77i)14-s + (10.1 − 12.3i)16-s − 13.9·17-s + (1.31 + 5.85i)18-s + 17.2i·19-s + ⋯ |
| L(s) = 1 | + (−0.219 − 0.975i)2-s − 0.577i·3-s + (−0.904 + 0.427i)4-s + (−0.563 + 0.126i)6-s − 0.904i·7-s + (0.615 + 0.788i)8-s − 0.333·9-s − 0.843i·11-s + (0.246 + 0.521i)12-s − 1.42·13-s + (−0.882 + 0.198i)14-s + (0.634 − 0.772i)16-s − 0.818·17-s + (0.0730 + 0.325i)18-s + 0.907i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.228509 + 0.360844i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228509 + 0.360844i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.438 + 1.95i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6.33iT - 49T^{2} \) |
| 11 | \( 1 + 9.27iT - 121T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 + 13.9T + 289T^{2} \) |
| 19 | \( 1 - 17.2iT - 361T^{2} \) |
| 23 | \( 1 - 33.7iT - 529T^{2} \) |
| 29 | \( 1 + 28.6T + 841T^{2} \) |
| 31 | \( 1 + 23.4iT - 961T^{2} \) |
| 37 | \( 1 - 67.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 50.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 81.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.49iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 13.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 69.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 46.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + 68.5T + 9.40e3T^{2} \) |
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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
−11.05032408145075675823717121958, −10.00891887278397266440056906839, −9.209529875083343198101283726732, −7.939925116088796939972537658787, −7.33460498762057531257913450559, −5.77148683274426778693862535034, −4.41440114105526530880037629608, −3.22034456893542871142340455196, −1.78888308364050783113450319094, −0.21645978723584261123595078948,
2.49715562258898346522697079458, 4.48679034289566324898923396244, 5.05247840425779705689903546308, 6.33639482463952426618490988933, 7.28141684721059552365725853693, 8.430947144183682287075866435448, 9.308375210149818725674327902524, 9.889578961440088575575220814029, 11.05611975239189760218148302095, 12.30863177140148080935132199447
