| L(s) = 1 | + (−0.457 + 2.59i)2-s + (−0.373 + 0.136i)3-s + (−4.64 − 1.69i)4-s + (−0.753 + 0.274i)5-s + (−0.181 − 1.03i)6-s + (−2.56 + 0.661i)7-s + (3.87 − 6.71i)8-s + (−2.17 + 1.82i)9-s + (−0.366 − 2.08i)10-s + 0.794·11-s + 1.96·12-s + (0.385 + 2.18i)13-s + (−0.544 − 6.95i)14-s + (0.244 − 0.204i)15-s + (8.07 + 6.77i)16-s + (4.47 + 3.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.323 + 1.83i)2-s + (−0.215 + 0.0785i)3-s + (−2.32 − 0.845i)4-s + (−0.336 + 0.122i)5-s + (−0.0742 − 0.421i)6-s + (−0.968 + 0.249i)7-s + (1.37 − 2.37i)8-s + (−0.725 + 0.608i)9-s + (−0.115 − 0.657i)10-s + 0.239·11-s + 0.567·12-s + (0.107 + 0.607i)13-s + (−0.145 − 1.85i)14-s + (0.0630 − 0.0529i)15-s + (2.01 + 1.69i)16-s + (1.08 + 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.174442 - 0.438253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174442 - 0.438253i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.56 - 0.661i)T \) |
| 19 | \( 1 + (-0.985 - 4.24i)T \) |
| good | 2 | \( 1 + (0.457 - 2.59i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.373 - 0.136i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (0.753 - 0.274i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 - 0.794T + 11T^{2} \) |
| 13 | \( 1 + (-0.385 - 2.18i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.47 - 3.75i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.811 + 4.60i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.60 + 1.67i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.68 - 2.91i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.63 + 2.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.47 - 8.38i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.05 - 2.56i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-9.41 + 7.89i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-8.03 - 2.92i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (9.18 + 7.70i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.77 - 10.0i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.747 + 4.24i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.94 - 4.14i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (5.76 - 2.09i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (6.51 + 5.46i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.76 - 6.51i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.73 - 3.17i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (3.07 - 1.11i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.28443116534629178834792599898, −13.27080352934836259200233332536, −12.05599715338864404960675901233, −10.42464603191703069528041805861, −9.386252839058647527248642192306, −8.367635121371662127043950865344, −7.45314044238030833693868886127, −6.21322013543095545621666553887, −5.56287440658120459306953908689, −3.90902154158138279507189639098,
0.56661962514623769807634493232, 2.91507075526951668272972265623, 3.83816282494232957328514639655, 5.58798453062699823560956993331, 7.53260634125055792959699864410, 9.018249866473875683229807451200, 9.620940761473463029562756471505, 10.70922944041928434354089522436, 11.71748704029651811270606136410, 12.24837153136002340293903208820
