| L(s) = 1 | + (−1.34 + 0.426i)2-s + (1.15 + 0.785i)3-s + (1.63 − 1.15i)4-s + (2.72 + 0.204i)5-s + (−1.88 − 0.567i)6-s + (1.69 − 2.03i)7-s + (−1.71 + 2.25i)8-s + (−0.386 − 0.983i)9-s + (−3.76 + 0.887i)10-s + (−4.00 − 1.57i)11-s + (2.78 − 0.0413i)12-s + (2.14 − 1.70i)13-s + (−1.41 + 3.46i)14-s + (2.97 + 2.37i)15-s + (1.35 − 3.76i)16-s + (2.62 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.953 + 0.301i)2-s + (0.664 + 0.453i)3-s + (0.817 − 0.575i)4-s + (1.21 + 0.0913i)5-s + (−0.770 − 0.231i)6-s + (0.640 − 0.768i)7-s + (−0.605 + 0.795i)8-s + (−0.128 − 0.327i)9-s + (−1.18 + 0.280i)10-s + (−1.20 − 0.474i)11-s + (0.804 − 0.0119i)12-s + (0.594 − 0.473i)13-s + (−0.378 + 0.925i)14-s + (0.768 + 0.613i)15-s + (0.337 − 0.941i)16-s + (0.635 + 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13516 + 0.185981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13516 + 0.185981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 - 0.426i)T \) |
| 7 | \( 1 + (-1.69 + 2.03i)T \) |
| good | 3 | \( 1 + (-1.15 - 0.785i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-2.72 - 0.204i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (4.00 + 1.57i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.14 + 1.70i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.62 - 2.82i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.33 - 5.76i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.71 - 4.00i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.688 - 3.01i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 2.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.82 + 2.41i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-3.34 - 6.95i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.60 + 3.33i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (8.45 - 1.27i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (8.59 - 2.65i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.523 + 6.98i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.0465 - 0.150i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-6.28 + 3.62i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.7 - 2.90i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.397 - 2.63i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-4.28 - 2.47i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.47 + 1.85i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (3.44 - 1.35i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 10.7iT - 97T^{2} \) |
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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
−12.61069586197004308011321101195, −11.00158274067778888908561962439, −10.26967001431667232176541417080, −9.744104094882741115522795110831, −8.389966657771255018863863669657, −7.958934181689399071784088440487, −6.31099622643405707335008354441, −5.44926291142565874538604655622, −3.37984377933486107273328195304, −1.72787103218514559869035677411,
2.00733081067223138209915545994, 2.57673962868198537766658221418, 5.11068839092889702697830013756, 6.42676232653323162294048156529, 7.74001139035220461694969941895, 8.513213972799904155874104592973, 9.344653753897819220604245821055, 10.33953073180870723404971832986, 11.27704952977229813320216646678, 12.48086585281368460459681533335
