| L(s) = 1 | + (−0.315 − 1.37i)2-s + (−1.04 − 1.38i)3-s + (−1.80 + 0.868i)4-s + (−0.437 + 0.159i)5-s + (−1.58 + 1.87i)6-s + (−3.46 + 0.610i)7-s + (1.76 + 2.21i)8-s + (−0.832 + 2.88i)9-s + (0.357 + 0.552i)10-s + (0.485 − 1.33i)11-s + (3.07 + 1.58i)12-s + (−1.62 + 1.93i)13-s + (1.93 + 4.58i)14-s + (0.675 + 0.439i)15-s + (2.49 − 3.12i)16-s + (−0.667 − 0.385i)17-s + ⋯ |
| L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.601 − 0.799i)3-s + (−0.900 + 0.434i)4-s + (−0.195 + 0.0711i)5-s + (−0.645 + 0.763i)6-s + (−1.30 + 0.230i)7-s + (0.624 + 0.781i)8-s + (−0.277 + 0.960i)9-s + (0.112 + 0.174i)10-s + (0.146 − 0.402i)11-s + (0.888 + 0.458i)12-s + (−0.449 + 0.535i)13-s + (0.516 + 1.22i)14-s + (0.174 + 0.113i)15-s + (0.622 − 0.782i)16-s + (−0.161 − 0.0934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0437270 + 0.0547507i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0437270 + 0.0547507i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.315 + 1.37i)T \) |
| 3 | \( 1 + (1.04 + 1.38i)T \) |
| good | 5 | \( 1 + (0.437 - 0.159i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.46 - 0.610i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.485 + 1.33i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.62 - 1.93i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.667 + 0.385i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.66 + 6.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.25 - 7.12i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.15 + 2.64i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (7.80 + 1.37i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (2.79 + 1.61i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.74 + 2.07i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.99 + 0.726i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.82 + 10.3i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 8.73T + 53T^{2} \) |
| 59 | \( 1 + (-0.818 - 2.24i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.02 + 0.533i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.04 + 3.39i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.70 + 9.88i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.96 - 5.12i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.94 - 10.6i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.41 - 8.83i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-8.59 + 4.96i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.95 + 0.710i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.62205295218147568760016856098, −10.95860574866028951655999538014, −9.714519811529002298268002092197, −8.924655907675245941669609681697, −7.55967042927320207955804337841, −6.53678149288749219158474006331, −5.23547472037929081746798398425, −3.61890738714297748119502077546, −2.19255419770559594894847813309, −0.06532386590356819459319581698,
3.64650718911709431732545408110, 4.68819608381465887637772923809, 6.03079060959760987471384497219, 6.63802957091113331749356129371, 8.028252497983500415665916906436, 9.189629063912286350978938960854, 10.06715111874053179667822190860, 10.57442003564406047781401664406, 12.36232468354787608773620493271, 12.79468617659515694128812201981
