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 − 0.999·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s − 3·11-s + (0.499 − 0.866i)12-s + (2.5 + 4.33i)13-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + (0.499 + 0.866i)18-s + (2.5 + 4.33i)19-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 − 0.353·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s − 0.904·11-s + (0.144 − 0.249i)12-s + (0.693 + 1.20i)13-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.242 + 0.420i)17-s + (0.117 + 0.204i)18-s + (0.573 + 0.993i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.955179215\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.955179215\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 6.06i)T \) |
good | 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2 + 3.46i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7 + 12.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.5 + 4.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−10.18292189491926584971762581825, −9.206007835969312755106643406564, −8.575555120512153594599415289318, −7.48300381395437310381146618550, −6.43070490361064058370796256610, −5.52512989908406569372306227121, −4.58560061388893452527437944368, −3.65289029493915044012674541124, −2.76778029490586732208383663897, −1.60514424115448976317171268497,
0.77068668204527063927287674008, 2.54712041799946166243148096999, 3.40622043192681713601759837762, 4.84736269019168766151916501055, 5.43264075326392101073570722249, 6.38661684263454426976783781788, 7.33284778826714492784211051714, 7.952092881369026163692617986427, 8.778575515497031681308082004788, 9.451527764073938227669520685484