L(s) = 1 | − 1.41·5-s + 2i·13-s + 1.41i·17-s + 1.00·25-s + 1.41·29-s + 1.41i·41-s − 49-s − 1.41·53-s − 2.82i·65-s − 2.00i·85-s − 1.41i·89-s + 1.41·101-s − 2i·109-s + 1.41i·113-s + ⋯ |
L(s) = 1 | − 1.41·5-s + 2i·13-s + 1.41i·17-s + 1.00·25-s + 1.41·29-s + 1.41i·41-s − 49-s − 1.41·53-s − 2.82i·65-s − 2.00i·85-s − 1.41i·89-s + 1.41·101-s − 2i·109-s + 1.41i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6888992838\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6888992838\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.22216496011576365503574831360, −9.245880145937487278194261597930, −8.417535917422872233665348684111, −7.85248963114647365968400683649, −6.82479710757392465753180640923, −6.24464345949396607014312935750, −4.65905070814679167101539162825, −4.19946092318241915619863649394, −3.21110176112132681611861235441, −1.66556699678536343319573575223,
0.64291267251029234845886976374, 2.80269654150583840512696627740, 3.49831560605953047502453314750, 4.65305696879996544277111566046, 5.39229759305115307216307119454, 6.62493620497504100776350074793, 7.59676607051189320959015277668, 7.982791495195756971958434785624, 8.835816872440099270693228017443, 9.913347031350007168799638457063