L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.714 − 1.23i)5-s + (2.10 + 1.59i)7-s + 0.999i·8-s + (1.23 − 0.714i)10-s + (−2.96 + 1.70i)11-s + 6.33i·13-s + (1.02 + 2.43i)14-s + (−0.5 + 0.866i)16-s + (−1.14 − 1.97i)17-s + (−1.87 − 1.08i)19-s + 1.42·20-s − 3.41·22-s + (6.97 + 4.02i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.319 − 0.553i)5-s + (0.797 + 0.603i)7-s + 0.353i·8-s + (0.391 − 0.226i)10-s + (−0.892 + 0.515i)11-s + 1.75i·13-s + (0.275 + 0.651i)14-s + (−0.125 + 0.216i)16-s + (−0.276 − 0.479i)17-s + (−0.430 − 0.248i)19-s + 0.319·20-s − 0.729·22-s + (1.45 + 0.839i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.480218835\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.480218835\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.10 - 1.59i)T \) |
good | 5 | \( 1 + (-0.714 + 1.23i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.96 - 1.70i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.33iT - 13T^{2} \) |
| 17 | \( 1 + (1.14 + 1.97i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.87 + 1.08i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.97 - 4.02i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.345iT - 29T^{2} \) |
| 31 | \( 1 + (-3.76 + 2.17i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.07 + 1.86i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.404T + 41T^{2} \) |
| 43 | \( 1 + 5.81T + 43T^{2} \) |
| 47 | \( 1 + (-2.75 + 4.77i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.56 + 4.94i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.51 + 9.55i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.94 - 5.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.12 + 3.68i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.55iT - 71T^{2} \) |
| 73 | \( 1 + (0.201 - 0.116i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.28 - 12.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.62T + 83T^{2} \) |
| 89 | \( 1 + (2.02 - 3.51i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−9.810142435756915890891133685814, −9.013259949828629302321263832006, −8.427118775990803060691043896244, −7.31685793032797969211842374969, −6.70802131943269556589176603919, −5.42973556995801875371260059434, −4.98270702865456352626905525201, −4.17483548204290258776697585017, −2.64369074809063038105543940534, −1.70864115957055169077426865473,
0.929051356261460369757154142425, 2.51661213806334950696614473367, 3.20917880285963060472759067117, 4.48896711936634848722722447613, 5.27188318930273238865717780272, 6.11432077943707475434492671431, 7.07810332047636938851315460160, 8.019551832395940891961274185593, 8.654968114077412397581611901769, 10.20220775192408686901070607383