L(s) = 1 | − i·3-s + (−1 − 2i)5-s + 4i·7-s − 9-s + 4i·13-s + (−2 + i)15-s + 8·19-s + 4·21-s + 4i·23-s + (−3 + 4i)25-s + i·27-s − 6·29-s + 8·31-s + (8 − 4i)35-s + 4i·37-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.447 − 0.894i)5-s + 1.51i·7-s − 0.333·9-s + 1.10i·13-s + (−0.516 + 0.258i)15-s + 1.83·19-s + 0.872·21-s + 0.834i·23-s + (−0.600 + 0.800i)25-s + 0.192i·27-s − 1.11·29-s + 1.43·31-s + (1.35 − 0.676i)35-s + 0.657i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31244 + 0.309826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31244 + 0.309826i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.650792749302337946181799221801, −9.262222206835457484755614124135, −8.415103588954964488254877030404, −7.71240976081536572422566947413, −6.70866737316784786177296608315, −5.60530398770157714774534466327, −5.09376829222111799629919852001, −3.75854924307519632440789484748, −2.48135447738237884783741304560, −1.29914658144467961439861519154,
0.71825506655189134218654800495, 2.85035100503354657465662311055, 3.62190216826083507701274624659, 4.46234821772170555779770077073, 5.59818820227675093164261046348, 6.68985947927623828947056456137, 7.58190292405612893024544673975, 7.968597947754405242181241195371, 9.401050543851123561989668694038, 10.15171153648370455755473970473