L(s) = 1 | + i·3-s + (−1.47 − 1.67i)5-s − 3.23i·7-s − 9-s + 1.20·11-s + 3.00i·13-s + (1.67 − 1.47i)15-s + 1.30i·17-s + 5.21·19-s + 3.23·21-s − 0.725i·23-s + (−0.636 + 4.95i)25-s − i·27-s + 6.60·29-s − 6.07·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.660 − 0.750i)5-s − 1.22i·7-s − 0.333·9-s + 0.364·11-s + 0.834i·13-s + (0.433 − 0.381i)15-s + 0.316i·17-s + 1.19·19-s + 0.705·21-s − 0.151i·23-s + (−0.127 + 0.991i)25-s − 0.192i·27-s + 1.22·29-s − 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.553995024\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.553995024\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.47 + 1.67i)T \) |
| 67 | \( 1 + iT \) |
good | 7 | \( 1 + 3.23iT - 7T^{2} \) |
| 11 | \( 1 - 1.20T + 11T^{2} \) |
| 13 | \( 1 - 3.00iT - 13T^{2} \) |
| 17 | \( 1 - 1.30iT - 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 + 0.725iT - 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 - 1.93iT - 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 + 8.35iT - 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 + 5.95iT - 53T^{2} \) |
| 59 | \( 1 + 0.552T + 59T^{2} \) |
| 61 | \( 1 + 5.13T + 61T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 9.22iT - 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 + 7.25iT - 83T^{2} \) |
| 89 | \( 1 - 5.48T + 89T^{2} \) |
| 97 | \( 1 + 2.07iT - 97T^{2} \) |
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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
−8.376654056965828673844397473644, −7.59015275075172794420505132834, −7.06631099480977123344340963688, −6.11691540059789001605585532241, −5.11243858019832332606207658756, −4.42747312467462570478002492521, −3.92119036156019838530463031283, −3.15466412134589215629406244934, −1.58859116284910655931060305194, −0.58994827405440233734528041599,
0.924063348768739960936110129662, 2.32314311021355373871697139443, 2.95111866199199036630739190526, 3.72062998639652970047763277253, 4.95713202035506618824512296119, 5.68806251115897650904427555430, 6.35072935817460115547330844482, 7.18778474482000671984934544686, 7.73957610664560745734165987551, 8.427548434232546441363979413106