L(s) = 1 | + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)9-s + (0.866 + 0.5i)13-s + (0.5 + 0.133i)17-s + (0.499 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (0.5 − 0.866i)37-s + (−0.5 + 0.133i)41-s − 0.999·45-s + (−0.5 + 0.866i)49-s + (1.36 − 1.36i)53-s + (−0.866 − 1.5i)61-s + (0.499 + 0.866i)65-s + 1.73i·73-s + (0.499 − 0.866i)81-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)9-s + (0.866 + 0.5i)13-s + (0.5 + 0.133i)17-s + (0.499 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (0.5 − 0.866i)37-s + (−0.5 + 0.133i)41-s − 0.999·45-s + (−0.5 + 0.866i)49-s + (1.36 − 1.36i)53-s + (−0.866 − 1.5i)61-s + (0.499 + 0.866i)65-s + 1.73i·73-s + (0.499 − 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.150134221\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.150134221\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
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
−10.18502039452881619749292887075, −9.413003214580630922561773697289, −8.617240154485949394044321898973, −7.74878916699767612711790689238, −6.70699298233330036146397907623, −5.90831070761394129172503879185, −5.27435569272871595675576351190, −3.89624204587248538763909709344, −2.81390136662141268355728347155, −1.75485612854312331570354177169,
1.25518480050533956782735392887, 2.69785545082404849515494256203, 3.72061460806248165640556200372, 5.05933640501617681941834643562, 5.79422500168449774282422454695, 6.42419058392455796541410810088, 7.65352467606666213179503770501, 8.638291678477199726445300089089, 9.085518839905576596524481069684, 10.02815004706293985733290622553