L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (3 − 5.19i)13-s + 0.999·15-s + 16-s + (0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (−2 − 3.46i)19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.150 − 0.261i)11-s + (−0.144 − 0.249i)12-s + (0.832 − 1.44i)13-s + 0.258·15-s + 0.250·16-s + (0.121 + 0.210i)17-s + (−0.117 + 0.204i)18-s + (−0.458 − 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03893 - 0.864751i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03893 - 0.864751i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-2 - 5.19i)T \) |
good | 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8 - 13.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.45078034470625065355973118830, −8.938021436634257134779824726231, −8.151449732357854426048694566240, −7.20575726398531714710647108545, −6.47854961586060048717879920478, −5.65169968243665552542141871900, −4.75764642334540718625258009648, −3.46724156998708297664841128708, −2.68314329674674070251577371413, −0.998065024629973811541892675766,
1.44378102724780495009563903983, 3.00785757185597485027405220046, 4.21290607583222055117696661273, 4.59919520368904898218184099730, 5.80717677167120521784200319955, 6.52919672020546172684816452615, 7.48284618114923384848553218494, 8.681136048549815276933184667397, 9.265983887851629742557743782328, 10.38358508955195293006385249304