L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 + 1.22i)5-s − i·7-s + (0.707 + 0.707i)8-s + (1.36 + 0.366i)10-s + 1.41i·11-s + (0.5 + 0.866i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + i·19-s − 1.41i·20-s + (1.36 − 0.366i)22-s + (1.22 − 0.707i)23-s + (−0.499 − 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 + 1.22i)5-s − i·7-s + (0.707 + 0.707i)8-s + (1.36 + 0.366i)10-s + 1.41i·11-s + (0.5 + 0.866i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + i·19-s − 1.41i·20-s + (1.36 − 0.366i)22-s + (1.22 − 0.707i)23-s + (−0.499 − 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6591954139\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6591954139\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−10.72252415070891778283177392164, −10.16554637694789441914139892782, −9.250207101813000458688940765732, −8.098863740985984264460307630708, −7.24269445120049559464084354034, −6.69674962109647417504543317936, −4.78459348626309175912940417740, −3.96614323793529465882074025846, −3.14317193113038057989371441228, −1.73012079626924733978588375793,
0.884902620780147459825508399959, 3.22219826610681404897451657560, 4.51204795151145310712983397807, 5.46673929003997776242664054341, 5.95467059176782995886409746803, 7.35989083935623609126236799932, 8.205071330055155422648068467838, 8.919544049587971227193325298360, 9.125310331037272911583546338928, 10.65432586193119046990098132776