L(s) = 1 | + (0.866 − 0.5i)2-s + (1.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (1.73 + 3i)5-s + 1.73·6-s + (−2.5 + 0.866i)7-s − 0.999i·8-s + (1.5 + 2.59i)9-s + (3 + 1.73i)10-s + (−2.59 − 1.5i)11-s + (1.49 − 0.866i)12-s + 3.46i·13-s + (−1.73 + 2i)14-s + 6i·15-s + (−0.5 − 0.866i)16-s + (0.866 − 1.5i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.866 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.774 + 1.34i)5-s + 0.707·6-s + (−0.944 + 0.327i)7-s − 0.353i·8-s + (0.5 + 0.866i)9-s + (0.948 + 0.547i)10-s + (−0.783 − 0.452i)11-s + (0.433 − 0.250i)12-s + 0.960i·13-s + (−0.462 + 0.534i)14-s + 1.54i·15-s + (−0.125 − 0.216i)16-s + (0.210 − 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.68108 + 1.43664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.68108 + 1.43664i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 + (-1.73 - 3i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-0.866 + 1.5i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (4.33 + 7.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.92 + 12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9iT - 71T^{2} \) |
| 73 | \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + (3.46 + 6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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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
−9.907280419103882669862681700751, −9.747491958664350789777048145926, −8.743576172040483476963821310737, −7.38024207767404975473705119753, −6.74322152754312084948694677583, −5.77893279322580307327085732682, −4.82840659399795029565920689336, −3.34615779536844302767168017299, −3.01422550492283049541736641957, −2.10168030921113068437880000780,
1.10342726057049501788041806710, 2.51616565001459956290848056573, 3.49900781072499852421896034421, 4.62873011110366232595882637017, 5.66176365005791203957000786204, 6.27069278489263540851888730706, 7.67584171890563718906824033329, 7.88138994095482360096174369650, 9.089661143606812505992883229061, 9.688283003029621609887291664318