L(s) = 1 | + (0.354 + 0.614i)3-s + (−0.5 − 0.866i)5-s − 3.11·7-s + (1.24 − 2.16i)9-s + 3.52·11-s + (−0.200 + 0.347i)13-s + (0.354 − 0.614i)15-s + (−1.74 − 3.02i)17-s + (−4.35 + 0.251i)19-s + (−1.10 − 1.91i)21-s + (−3.65 + 6.33i)23-s + (−0.499 + 0.866i)25-s + 3.89·27-s + (3.96 − 6.86i)29-s − 5.73·31-s + ⋯ |
L(s) = 1 | + (0.204 + 0.354i)3-s + (−0.223 − 0.387i)5-s − 1.17·7-s + (0.416 − 0.720i)9-s + 1.06·11-s + (−0.0556 + 0.0964i)13-s + (0.0915 − 0.158i)15-s + (−0.424 − 0.734i)17-s + (−0.998 + 0.0577i)19-s + (−0.240 − 0.416i)21-s + (−0.762 + 1.32i)23-s + (−0.0999 + 0.173i)25-s + 0.750·27-s + (0.736 − 1.27i)29-s − 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7202629017\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7202629017\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (4.35 - 0.251i)T \) |
good | 3 | \( 1 + (-0.354 - 0.614i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 + (0.200 - 0.347i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.74 + 3.02i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.65 - 6.33i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.96 + 6.86i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + (-0.555 - 0.961i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.30 + 7.45i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.76 + 6.51i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.27 - 9.14i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.25 + 7.37i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.61 + 7.98i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.20 + 7.28i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.31 + 7.46i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.870 + 1.50i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.50 + 7.80i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.07T + 83T^{2} \) |
| 89 | \( 1 + (-6.61 + 11.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.85 - 6.67i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.224128536525669568464742311816, −8.706603105679073087078924635152, −7.48630990734245394677953255227, −6.64865981875774277057350028111, −6.11920566467127617687515026551, −4.85217017204863463020469629143, −3.83689208774758783939074639033, −3.44226504253301436005943124126, −1.88605308473779344674031651835, −0.26609838781097338435771723128,
1.63327601124183956157899777640, 2.73922715050870719954478993093, 3.76903206892149917508099536114, 4.55045968658114508315576856917, 5.92082989585198262758775680186, 6.76717869731185965756418655495, 7.00949244189798192937279263720, 8.334791056054143911756551547762, 8.757645058880751564778721898908, 9.856688763666385647000287001259