L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1.5 − 2.59i)5-s − 0.999·6-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + 6·13-s + (−0.500 − 2.59i)14-s + 3·15-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s − 0.408·6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + (0.150 − 0.261i)11-s + (−0.144 − 0.249i)12-s + 1.66·13-s + (−0.133 − 0.694i)14-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.819390 - 0.406194i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.819390 - 0.406194i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.5 + 4.33i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−11.08596382079880848787592403646, −9.872543175763159238176105725147, −8.866691796209111222041956602010, −8.415653659651287878995620351852, −7.07203850182775116138844693569, −6.17017495147869987393925431088, −5.13683434004154946567090740583, −4.17754879571862179316054462549, −3.36858210766550814086785619680, −0.53287437485740683105990951494,
1.80139740433306722210677912276, 3.41053038581358369306414975653, 3.82307835903289348310255895215, 5.93771831703104399924578753623, 6.19563026165788213932908796184, 7.44274598769626041223788770633, 8.401261182946546531249815981879, 9.677191152823600531507542156007, 10.57563800529633923998914339475, 11.20563809397101014650664970573