| L(s) = 1 | + (−0.654 − 0.755i)3-s + (0.217 + 1.51i)5-s + (−1.55 + 3.40i)7-s + (−0.142 + 0.989i)9-s + (1.04 + 0.306i)11-s + (1.25 + 2.75i)13-s + (0.999 − 1.15i)15-s + (2.61 + 1.67i)17-s + (−3.49 + 2.24i)19-s + (3.59 − 1.05i)21-s + (2.35 − 4.17i)23-s + (2.56 − 0.752i)25-s + (0.841 − 0.540i)27-s + (2.24 + 1.44i)29-s + (−4.45 + 5.14i)31-s + ⋯ |
| L(s) = 1 | + (−0.378 − 0.436i)3-s + (0.0971 + 0.675i)5-s + (−0.588 + 1.28i)7-s + (−0.0474 + 0.329i)9-s + (0.314 + 0.0924i)11-s + (0.348 + 0.763i)13-s + (0.258 − 0.297i)15-s + (0.633 + 0.406i)17-s + (−0.801 + 0.514i)19-s + (0.784 − 0.230i)21-s + (0.490 − 0.871i)23-s + (0.512 − 0.150i)25-s + (0.161 − 0.104i)27-s + (0.416 + 0.267i)29-s + (−0.800 + 0.923i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.844931 + 0.564303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.844931 + 0.564303i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-2.35 + 4.17i)T \) |
| good | 5 | \( 1 + (-0.217 - 1.51i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.55 - 3.40i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.04 - 0.306i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 2.75i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.61 - 1.67i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.49 - 2.24i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.24 - 1.44i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.45 - 5.14i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.973 - 6.77i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (1.41 + 9.85i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.62 + 3.02i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 + (1.19 - 2.61i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (6.11 + 13.3i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.42 + 3.95i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.44 + 1.30i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-14.6 + 4.29i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.70 + 1.73i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.28 + 2.81i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.773 - 5.38i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-9.86 - 11.3i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.546 - 3.79i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.36948760051108427296643825222, −11.14289667133828087845626589694, −10.31472786000520518068121516855, −9.123699360693503082081489988281, −8.313389210831905279593275706341, −6.74855788688617894579786066020, −6.35311559213067962005021197255, −5.13567184016359121321905549635, −3.40248283075114421657503639284, −2.04343498870579159322690080433,
0.853290761524924203036743056111, 3.35679416204171119199026360836, 4.42320773213047918543920054179, 5.55329969739753103366606864152, 6.71879737109821401466870865570, 7.78206979281857942058673731699, 9.036975281482694232316690030518, 9.873439618699385232466704211674, 10.71848031150725793680178702685, 11.55888613637923699589074201941
