| L(s) = 1 | + (−1.32 + 0.492i)2-s + (−1.28 − 1.16i)3-s + (1.51 − 1.30i)4-s + (−2.03 − 2.64i)5-s + (2.27 + 0.910i)6-s + (1.18 − 4.43i)7-s + (−1.36 + 2.47i)8-s + (0.293 + 2.98i)9-s + (3.99 + 2.51i)10-s + (0.532 + 4.04i)11-s + (−3.46 − 0.0888i)12-s + (−4.53 − 0.596i)13-s + (0.607 + 6.47i)14-s + (−0.473 + 5.76i)15-s + (0.594 − 3.95i)16-s − 2.19i·17-s + ⋯ |
| L(s) = 1 | + (−0.937 + 0.347i)2-s + (−0.740 − 0.671i)3-s + (0.757 − 0.652i)4-s + (−0.909 − 1.18i)5-s + (0.928 + 0.371i)6-s + (0.449 − 1.67i)7-s + (−0.483 + 0.875i)8-s + (0.0977 + 0.995i)9-s + (1.26 + 0.794i)10-s + (0.160 + 1.21i)11-s + (−0.999 − 0.0256i)12-s + (−1.25 − 0.165i)13-s + (0.162 + 1.72i)14-s + (−0.122 + 1.48i)15-s + (0.148 − 0.988i)16-s − 0.532i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0153250 + 0.277230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0153250 + 0.277230i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 - 0.492i)T \) |
| 3 | \( 1 + (1.28 + 1.16i)T \) |
| good | 5 | \( 1 + (2.03 + 2.64i)T + (-1.29 + 4.82i)T^{2} \) |
| 7 | \( 1 + (-1.18 + 4.43i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.532 - 4.04i)T + (-10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (4.53 + 0.596i)T + (12.5 + 3.36i)T^{2} \) |
| 17 | \( 1 + 2.19iT - 17T^{2} \) |
| 19 | \( 1 + (-0.968 - 2.33i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.0965 - 0.360i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (5.83 + 4.47i)T + (7.50 + 28.0i)T^{2} \) |
| 31 | \( 1 + (1.48 - 2.56i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.05 - 2.53i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.754 + 2.81i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.00 - 7.60i)T + (-41.5 + 11.1i)T^{2} \) |
| 47 | \( 1 + (-3.38 + 1.95i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.97 + 2.47i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (2.52 + 3.28i)T + (-15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-5.29 - 4.06i)T + (15.7 + 58.9i)T^{2} \) |
| 67 | \( 1 + (-1.11 + 8.45i)T + (-64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (-0.270 - 0.270i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.07 - 3.07i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.50 + 3.18i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.12 + 9.28i)T + (-21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (2.55 + 2.55i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.02 + 6.97i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.36854329398387287296008715256, −10.32589048697820889499978724086, −9.505425213931534482000389888964, −7.949715894095580027411211101827, −7.57921759799552867157712639199, −6.89749200667539537265948605857, −5.18004992605421365023162589562, −4.42227263082345041600858411570, −1.60108079395144679585700305072, −0.30727111951441245554472422680,
2.59022078564222748023074724758, 3.66723892720385695385754458285, 5.42270924262942147819050433499, 6.49588922898548268162661371107, 7.57709104932152570497253444038, 8.716946691114371282621471497859, 9.457904729240360484292465638847, 10.70434476208612833038850152304, 11.23909612375998605080681641465, 11.83366707365713006201214300559
