| L(s) = 1 | + (−1.35 − 0.418i)2-s + (−0.991 − 0.130i)3-s + (1.64 + 1.13i)4-s + (−0.308 − 2.34i)5-s + (1.28 + 0.591i)6-s + (−2.03 − 1.69i)7-s + (−1.75 − 2.21i)8-s + (0.965 + 0.258i)9-s + (−0.564 + 3.29i)10-s + (−0.192 − 0.250i)11-s + (−1.48 − 1.33i)12-s + (−2.68 + 1.11i)13-s + (2.04 + 3.13i)14-s + 2.36i·15-s + (1.43 + 3.73i)16-s + (−6.06 − 3.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.955 − 0.296i)2-s + (−0.572 − 0.0753i)3-s + (0.824 + 0.565i)4-s + (−0.137 − 1.04i)5-s + (0.524 + 0.241i)6-s + (−0.769 − 0.638i)7-s + (−0.619 − 0.784i)8-s + (0.321 + 0.0862i)9-s + (−0.178 + 1.04i)10-s + (−0.0580 − 0.0755i)11-s + (−0.429 − 0.385i)12-s + (−0.743 + 0.308i)13-s + (0.545 + 0.837i)14-s + 0.609i·15-s + (0.359 + 0.933i)16-s + (−1.46 − 0.848i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00355209 + 0.00549894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00355209 + 0.00549894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 + 0.418i)T \) |
| 3 | \( 1 + (0.991 + 0.130i)T \) |
| 7 | \( 1 + (2.03 + 1.69i)T \) |
| good | 5 | \( 1 + (0.308 + 2.34i)T + (-4.82 + 1.29i)T^{2} \) |
| 11 | \( 1 + (0.192 + 0.250i)T + (-2.84 + 10.6i)T^{2} \) |
| 13 | \( 1 + (2.68 - 1.11i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (6.06 + 3.49i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.94 - 1.49i)T + (4.91 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-8.20 - 2.19i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.81 + 4.38i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (4.88 - 8.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.565 + 4.29i)T + (-35.7 + 9.57i)T^{2} \) |
| 41 | \( 1 + (4.89 - 4.89i)T - 41iT^{2} \) |
| 43 | \( 1 + (3.77 - 9.12i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-2.17 + 1.25i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.998 + 1.30i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (3.86 - 2.96i)T + (15.2 - 56.9i)T^{2} \) |
| 61 | \( 1 + (-2.61 + 3.40i)T + (-15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (8.00 + 1.05i)T + (64.7 + 17.3i)T^{2} \) |
| 71 | \( 1 + (-6.21 - 6.21i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.04 - 11.3i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.72 - 3.88i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.82 - 2.41i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (2.72 - 10.1i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 6.05T + 97T^{2} \) |
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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
−9.671718624189928148114348015539, −9.300714153883669875459609188888, −8.344931230214761540696738092587, −7.12661072191639132024282314641, −6.80248248473967434700231080853, −5.32742109678275062360151749406, −4.31936128623664639492183359530, −2.93745796954910311245035711964, −1.28557462646900429715923889376, −0.00532078601860305866603200792,
2.21379248243841110526486348403, 3.24781487097268265661993461217, 5.03372085241489655616137705559, 6.04918960609670078521843746370, 6.87022763605402667340085670937, 7.28915347344478601923158149722, 8.707726559775193109070553713524, 9.330548859226214796326562134340, 10.34749849146184778891992882574, 10.84389468132036310349395499027
