| L(s) = 1 | + (−1.57 + 0.726i)3-s + (−1.78 + 0.651i)5-s + (−0.623 − 3.53i)7-s + (1.94 − 2.28i)9-s + (−4.04 − 1.47i)11-s + (−0.0714 − 0.0599i)13-s + (2.34 − 2.32i)15-s + (1.83 − 3.18i)17-s + (−3.88 − 6.72i)19-s + (3.54 + 5.10i)21-s + (−1.14 + 6.49i)23-s + (−1.05 + 0.883i)25-s + (−1.39 + 5.00i)27-s + (−3.61 + 3.03i)29-s + (−0.0708 + 0.401i)31-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.419i)3-s + (−0.800 + 0.291i)5-s + (−0.235 − 1.33i)7-s + (0.648 − 0.761i)9-s + (−1.22 − 0.444i)11-s + (−0.0198 − 0.0166i)13-s + (0.604 − 0.599i)15-s + (0.446 − 0.772i)17-s + (−0.890 − 1.54i)19-s + (0.774 + 1.11i)21-s + (−0.238 + 1.35i)23-s + (−0.210 + 0.176i)25-s + (−0.269 + 0.963i)27-s + (−0.671 + 0.563i)29-s + (−0.0127 + 0.0721i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.172747 - 0.313449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.172747 - 0.313449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 - 0.726i)T \) |
| good | 5 | \( 1 + (1.78 - 0.651i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.623 + 3.53i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (4.04 + 1.47i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0714 + 0.0599i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.83 + 3.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.88 + 6.72i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.14 - 6.49i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (3.61 - 3.03i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.0708 - 0.401i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.11 + 3.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.23 + 4.39i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.2 - 3.74i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.472 + 2.67i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 + (-1.48 + 0.539i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.39 - 7.93i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.15 + 6.00i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.22 - 3.84i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.95 + 8.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.33 + 3.64i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.85 + 4.91i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (5.55 + 9.61i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.0 - 4.73i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.69929999852489808995850885287, −10.95452398462674156163924108610, −10.41342486229685878072431208794, −9.244739249793444096035477284410, −7.57971888905788345584748049041, −7.09580043752419026143143603551, −5.60894595604664644312304603594, −4.45297643259879963845243292481, −3.36240500316061556007814602720, −0.32327273344146146011269907839,
2.24625830657011086160804694172, 4.25330536942561555542306842547, 5.51432041924180478309433179583, 6.28493723952091946153254849626, 7.82031787824093306885993513918, 8.356872709188147838709086987152, 9.960222035815164132546186346538, 10.79799580145559741267822536386, 12.04158591442983314951448248481, 12.38670334215027500384078488153
