L(s) = 1 | + (−1.22 − 0.707i)2-s + (−0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (1.30 − 0.541i)6-s + (−0.258 − 0.965i)9-s + (−0.991 − 0.130i)12-s − 1.84i·13-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)18-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)25-s + (−1.30 + 2.26i)26-s + (0.923 + 0.382i)27-s + (−0.662 + 0.382i)31-s + (−1.22 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (−1.22 − 0.707i)2-s + (−0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (1.30 − 0.541i)6-s + (−0.258 − 0.965i)9-s + (−0.991 − 0.130i)12-s − 1.84i·13-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)18-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)25-s + (−1.30 + 2.26i)26-s + (0.923 + 0.382i)27-s + (−0.662 + 0.382i)31-s + (−1.22 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2464455454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2464455454\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.608 - 0.793i)T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
good | 2 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 1.84iT - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−8.606269585929822481257381974774, −8.092779238995882072689545553101, −7.24682534476405637172279798381, −6.15668805560165558483150224699, −5.38412889746415941784334810275, −4.76368769045129678946519317422, −3.42318472562653766491916802414, −2.91307453218250301166296148399, −1.47959158568745335716763277516, −0.26188183844456473860922889908,
1.31857315506665504231014701508, 2.10049388927937453366403394985, 3.65179052307374927676773675046, 4.74918498697115531879625764567, 5.65290351844305126627005469202, 6.54084216721992246152357262898, 6.95747574559628553497284031416, 7.44456808327849142070361756144, 8.417254175395056919415743219143, 8.849184246166155498798356869921