L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s − i·7-s + (−0.707 + 0.707i)8-s − 0.517·11-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (0.499 − 0.133i)22-s + 1.41·23-s − 25-s + (−0.499 − 0.866i)28-s − 1.41i·29-s + (−0.258 + 0.965i)32-s + 1.73·37-s − i·43-s + (−0.448 + 0.258i)44-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s − i·7-s + (−0.707 + 0.707i)8-s − 0.517·11-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (0.499 − 0.133i)22-s + 1.41·23-s − 25-s + (−0.499 − 0.866i)28-s − 1.41i·29-s + (−0.258 + 0.965i)32-s + 1.73·37-s − i·43-s + (−0.448 + 0.258i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6776734196\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6776734196\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + 0.517T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - 1.73T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.93iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 + 1.93T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.73iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.131145678906022624079289313854, −8.150540743898908832915529916411, −7.67487646132822671820001499438, −6.95325736985320271669209316228, −6.18029093126702965573749876514, −5.27895468684522322010884974587, −4.25377382597382936665630537466, −3.10523523773104528916494398548, −2.01096185849883498878181829577, −0.67428576824469390893144468412,
1.36745054032952608524540755017, 2.56571707932231761732010390153, 3.15192038663626936733336215520, 4.52332853197185583558295428209, 5.61331157669235624663230170093, 6.27349546219334578834723193992, 7.29766735729493335593525855958, 7.86163230470940582080745008716, 8.815727586821140923094503940490, 9.146365789043012499893729696002