L(s) = 1 | + (−1.69 − 1.45i)5-s + (−2.91 − 2.91i)7-s + 1.52i·11-s + (−3.25 + 3.25i)13-s + (−3.16 + 3.16i)17-s − 2.06i·19-s + (−0.539 − 0.539i)23-s + (0.754 + 4.94i)25-s + 9.58·29-s + 0.803·31-s + (0.697 + 9.19i)35-s + (−3.32 − 3.32i)37-s + 4.89i·41-s + (0.0464 − 0.0464i)43-s + (7.02 − 7.02i)47-s + ⋯ |
L(s) = 1 | + (−0.758 − 0.651i)5-s + (−1.10 − 1.10i)7-s + 0.461i·11-s + (−0.901 + 0.901i)13-s + (−0.767 + 0.767i)17-s − 0.473i·19-s + (−0.112 − 0.112i)23-s + (0.150 + 0.988i)25-s + 1.78·29-s + 0.144·31-s + (0.117 + 1.55i)35-s + (−0.547 − 0.547i)37-s + 0.764i·41-s + (0.00708 − 0.00708i)43-s + (1.02 − 1.02i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7868151787\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7868151787\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.69 + 1.45i)T \) |
good | 7 | \( 1 + (2.91 + 2.91i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.52iT - 11T^{2} \) |
| 13 | \( 1 + (3.25 - 3.25i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.16 - 3.16i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.06iT - 19T^{2} \) |
| 23 | \( 1 + (0.539 + 0.539i)T + 23iT^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 31 | \( 1 - 0.803T + 31T^{2} \) |
| 37 | \( 1 + (3.32 + 3.32i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.89iT - 41T^{2} \) |
| 43 | \( 1 + (-0.0464 + 0.0464i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.02 + 7.02i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.61 - 7.61i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.23T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 + (11.0 + 11.0i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.107iT - 71T^{2} \) |
| 73 | \( 1 + (4.71 - 4.71i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 + (-7.89 - 7.89i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.07T + 89T^{2} \) |
| 97 | \( 1 + (-6.71 - 6.71i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.075042434526323589933198627201, −8.495446499704557132989588170116, −7.36648932536298473990617653271, −7.00829958596015817562648529464, −6.21162843467317270301487787331, −4.82996533312532524302107308458, −4.31204841094821516712838350834, −3.57220253521972574562999664194, −2.32648046722469680600203210045, −0.814135945747217337802104771348,
0.38351809432283431126016301306, 2.60026303810385941120725877278, 2.88683460279386117490275368451, 3.95307940759861090800958941159, 5.08506000821612651410503528904, 5.91173518474648509936215270948, 6.70861192105462211270867702740, 7.33458060382052079750266801981, 8.356694186373997608298797785275, 8.842139163589524358459167028001