L(s) = 1 | + i·2-s − 4-s − i·8-s − 9-s − 11-s + i·13-s + 16-s − i·18-s − 19-s − i·22-s − i·23-s − 26-s − 29-s − 31-s + i·32-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − i·8-s − 9-s − 11-s + i·13-s + 16-s − i·18-s − 19-s − i·22-s − i·23-s − 26-s − 29-s − 31-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07109754748\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07109754748\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + 2iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - iT - T^{2} \) |
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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
−9.447181547044327563422811181416, −8.810010100406638396570258736277, −8.222741049483114204814300005896, −7.45752204034162704565941380819, −6.59454571379824689759498745273, −5.94037398288897289281797444252, −5.12945211479962362694841121928, −4.36727935623924144072347413503, −3.32597661686900365821911414891, −2.09836532112189650068127205079,
0.04592808025346633461755157435, 1.81312301984383292539524334447, 2.84533679984173655368431239822, 3.46427667840631677358189091983, 4.62678569208065006604764918821, 5.49636008104386237177672134596, 5.97296086500128917567425007368, 7.55561991901773944070601556894, 8.028629323808966637283428892451, 8.898108053760377077895454950228