| L(s) = 1 | + (1.39 + 0.261i)2-s + (0.720 + 1.44i)3-s + (0.0247 + 0.00958i)4-s + (−3.44 + 2.13i)5-s + (0.629 + 2.21i)6-s + (−1.57 + 2.08i)7-s + (−2.38 − 1.47i)8-s + (0.233 − 0.309i)9-s + (−5.38 + 2.08i)10-s + (5.41 + 2.09i)11-s + (0.00395 + 0.0426i)12-s + (2.50 + 1.55i)13-s + (−2.75 + 2.51i)14-s + (−5.56 − 3.44i)15-s + (−2.99 − 2.73i)16-s + (2.34 + 3.39i)17-s + ⋯ |
| L(s) = 1 | + (0.989 + 0.184i)2-s + (0.415 + 0.835i)3-s + (0.0123 + 0.00479i)4-s + (−1.54 + 0.954i)5-s + (0.257 + 0.903i)6-s + (−0.596 + 0.789i)7-s + (−0.844 − 0.522i)8-s + (0.0778 − 0.103i)9-s + (−1.70 + 0.659i)10-s + (1.63 + 0.632i)11-s + (0.00114 + 0.0123i)12-s + (0.695 + 0.430i)13-s + (−0.735 + 0.670i)14-s + (−1.43 − 0.890i)15-s + (−0.748 − 0.682i)16-s + (0.569 + 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.847651 + 1.37536i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.847651 + 1.37536i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-2.34 - 3.39i)T \) |
| good | 2 | \( 1 + (-1.39 - 0.261i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (-0.720 - 1.44i)T + (-1.80 + 2.39i)T^{2} \) |
| 5 | \( 1 + (3.44 - 2.13i)T + (2.22 - 4.47i)T^{2} \) |
| 7 | \( 1 + (1.57 - 2.08i)T + (-1.91 - 6.73i)T^{2} \) |
| 11 | \( 1 + (-5.41 - 2.09i)T + (8.12 + 7.41i)T^{2} \) |
| 13 | \( 1 + (-2.50 - 1.55i)T + (5.79 + 11.6i)T^{2} \) |
| 19 | \( 1 + (3.54 - 0.663i)T + (17.7 - 6.86i)T^{2} \) |
| 23 | \( 1 + (-4.38 + 5.80i)T + (-6.29 - 22.1i)T^{2} \) |
| 29 | \( 1 + (6.39 + 2.47i)T + (21.4 + 19.5i)T^{2} \) |
| 31 | \( 1 + (-2.79 - 1.73i)T + (13.8 + 27.7i)T^{2} \) |
| 37 | \( 1 + (0.223 - 2.41i)T + (-36.3 - 6.79i)T^{2} \) |
| 41 | \( 1 + (-0.376 - 0.756i)T + (-24.7 + 32.7i)T^{2} \) |
| 43 | \( 1 + (4.24 - 3.87i)T + (3.96 - 42.8i)T^{2} \) |
| 47 | \( 1 + (3.09 + 4.09i)T + (-12.8 + 45.2i)T^{2} \) |
| 53 | \( 1 + (0.533 - 0.706i)T + (-14.5 - 50.9i)T^{2} \) |
| 59 | \( 1 + (-0.837 + 2.94i)T + (-50.1 - 31.0i)T^{2} \) |
| 61 | \( 1 + (-2.51 - 8.82i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (-11.8 + 2.22i)T + (62.4 - 24.2i)T^{2} \) |
| 71 | \( 1 + (-1.95 + 2.59i)T + (-19.4 - 68.2i)T^{2} \) |
| 73 | \( 1 + (2.87 + 2.61i)T + (6.73 + 72.6i)T^{2} \) |
| 79 | \( 1 + (4.64 - 0.867i)T + (73.6 - 28.5i)T^{2} \) |
| 83 | \( 1 + (-2.24 + 4.50i)T + (-50.0 - 66.2i)T^{2} \) |
| 89 | \( 1 + (-8.57 + 5.30i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (-9.44 + 12.5i)T + (-26.5 - 93.2i)T^{2} \) |
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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
−12.17727783621899539740021753146, −11.44408582744692708419169967184, −10.23389913456011003037171575243, −9.204943354664502188927264581319, −8.484487596242344864131174482583, −6.78325252590842224307085892158, −6.32141170750302848355962573128, −4.45114906660606921610158756884, −3.88136021829199281816116746700, −3.18612858048081628815493497200,
0.950131372572710938558229288070, 3.47801727849953681982389315944, 3.88223599682640474396455480383, 5.12687978746748921853450217306, 6.64215513177120402072481896057, 7.59875015142694449911633023992, 8.525212973303472327304860838174, 9.261354282565103173401887780959, 11.17455442933868444384305046693, 11.75713040849107959570830172934
