L(s) = 1 | + 4-s − 1.41i·7-s + 1.41i·13-s + 16-s − 1.41i·19-s − 25-s − 1.41i·28-s + 1.41i·43-s − 1.00·49-s + 1.41i·52-s + 1.41i·61-s + 64-s + 1.41i·73-s − 1.41i·76-s − 1.41i·79-s + ⋯ |
L(s) = 1 | + 4-s − 1.41i·7-s + 1.41i·13-s + 16-s − 1.41i·19-s − 25-s − 1.41i·28-s + 1.41i·43-s − 1.00·49-s + 1.41i·52-s + 1.41i·61-s + 64-s + 1.41i·73-s − 1.41i·76-s − 1.41i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.286727564\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.286727564\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−10.06721808329148398180358714857, −9.355268859818653488796930526917, −8.201978185270831678360105863139, −7.21026740810000805014704578668, −6.92379717397623357864914329454, −5.99374002314788415982377879248, −4.63826198419024545315704109965, −3.85803134120321147141647257358, −2.62409618595247610835911859261, −1.39050397970367248922835682035,
1.80435812220385207565973103636, 2.74495422624022727431092388158, 3.68341013037308675448901994764, 5.44367271914378711259131893485, 5.72178608052953435384779413066, 6.66798072466537257219409274280, 7.86999774936843300799674658882, 8.210923552930852750726430637685, 9.375810229740356501027215040802, 10.21096728128016961756596708127