L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.330 + 0.571i)5-s + (−2.34 + 1.21i)7-s + 0.999i·8-s + (−0.571 − 0.330i)10-s + (2.25 + 1.30i)11-s − 5.87i·13-s + (1.42 − 2.22i)14-s + (−0.5 − 0.866i)16-s + (2.35 − 4.08i)17-s + (−3.59 + 2.07i)19-s + 0.660·20-s − 2.60·22-s + (4.56 − 2.63i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.147 + 0.255i)5-s + (−0.888 + 0.459i)7-s + 0.353i·8-s + (−0.180 − 0.104i)10-s + (0.678 + 0.392i)11-s − 1.62i·13-s + (0.381 − 0.595i)14-s + (−0.125 − 0.216i)16-s + (0.571 − 0.989i)17-s + (−0.824 + 0.475i)19-s + 0.147·20-s − 0.554·22-s + (0.951 − 0.549i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.116272309\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.116272309\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.34 - 1.21i)T \) |
good | 5 | \( 1 + (-0.330 - 0.571i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.25 - 1.30i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.87iT - 13T^{2} \) |
| 17 | \( 1 + (-2.35 + 4.08i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.59 - 2.07i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.56 + 2.63i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.43iT - 29T^{2} \) |
| 31 | \( 1 + (1.73 + 1.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.17 - 5.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.90T + 41T^{2} \) |
| 43 | \( 1 + 2.66T + 43T^{2} \) |
| 47 | \( 1 + (-0.874 - 1.51i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.83 - 4.52i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.111 - 0.193i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.78 + 4.49i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.67 + 9.82i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.89iT - 71T^{2} \) |
| 73 | \( 1 + (3.46 + 2.00i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.08 - 8.80i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + (7.52 + 13.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.33iT - 97T^{2} \) |
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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
−9.791776646537198434447567296329, −9.027508716967403424738195927817, −8.272516675704569624583867273815, −7.29026604548689179616978044995, −6.57001395742021816703561006594, −5.80537921138023231813012759520, −4.87127136667372294822454590901, −3.35531833977992055940259775212, −2.51973931061024706715730234955, −0.790443847870108281787902039344,
1.00601636776239015735004265159, 2.26684409219741247989526825242, 3.64270221577690257635314592658, 4.24247368923131037904995220698, 5.76922418414892560854985157039, 6.65680052749045254675318708001, 7.23356515840505998496970990881, 8.427191875300766837292298978383, 9.203676264207622465676151956320, 9.566545718021393099290677548602