L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·6-s + (2 + 3.46i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s + 3·11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s − 3.99·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + 0.408·6-s + (0.755 + 1.30i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s + 0.904·11-s + (−0.144 + 0.249i)12-s + (0.138 + 0.240i)13-s − 1.06·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.255512523\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255512523\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-5.5 - 2.59i)T \) |
good | 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.878239054423569226636079936166, −8.999582932397830252761232864753, −8.508513689942342705592719183176, −7.56782466759933503910621492650, −6.65102638732774285269019993368, −6.00643948418209814195783956982, −5.27649957827327823292054789467, −4.12805427966953877989478370948, −2.46758968551662027939843231709, −1.49088098446221964131153324016,
0.69945685765039033126907425969, 1.84743324634711795347959393023, 3.51204786983622105353654680162, 4.26007425008852788319433041974, 4.99890206292190340404723554584, 6.21966732912253373981346406212, 7.29661070892775548685309680603, 8.051834098891151457851135003076, 9.065505156518732485211837234604, 9.633258673698703187860387889141