L(s) = 1 | + (−1.5 + 0.866i)3-s + (−1 + 1.73i)5-s + (−0.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + (0.5 + 0.866i)11-s + (3 − 5.19i)13-s − 3.46i·15-s − 5·17-s + 7·19-s + (1.5 + 0.866i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 5.19i·27-s + (2 + 3.46i)29-s + (−3 + 5.19i)31-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.188 − 0.327i)7-s + (0.5 − 0.866i)9-s + (0.150 + 0.261i)11-s + (0.832 − 1.44i)13-s − 0.894i·15-s − 1.21·17-s + 1.60·19-s + (0.327 + 0.188i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.999i·27-s + (0.371 + 0.643i)29-s + (−0.538 + 0.933i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.066941349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066941349\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 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 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8 - 13.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.36888140826630746176488953348, −9.390380442065762142082684866181, −8.437375492676719228509422488476, −7.23215565870312661102432643279, −6.77451638415183494774830382539, −5.72133422824476750017118463218, −4.88781362742784228309991891096, −3.74294154192593674966189724795, −3.02128274132811354066849985770, −0.906756118738555286380305438060,
0.812235298555252510886035275990, 2.07452330142743886164965910853, 3.80716737012656742753774995804, 4.69645324057902840504579966857, 5.59414197443045563636132795290, 6.47773240772197719564003062171, 7.21114181375660063017265460879, 8.232515500873521451287137570030, 9.024756521375990362504658290706, 9.750602134832573164493971686400