| L(s) = 1 | + (2 − i)5-s − 2i·7-s + 6·11-s − 2i·13-s − 6i·17-s − 4·19-s + 8i·23-s + (3 − 4i)25-s − 8·31-s + (−2 − 4i)35-s − 2i·37-s + 6·41-s + 4i·43-s − 4i·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + (0.894 − 0.447i)5-s − 0.755i·7-s + 1.80·11-s − 0.554i·13-s − 1.45i·17-s − 0.917·19-s + 1.66i·23-s + (0.600 − 0.800i)25-s − 1.43·31-s + (−0.338 − 0.676i)35-s − 0.328i·37-s + 0.937·41-s + 0.609i·43-s − 0.583i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.096746509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.096746509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 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
−9.317303426941891077866547460954, −8.922545014685414204835721066245, −7.64935600880662294895007811941, −6.94645429673339141052476489839, −6.10493581300060518518083163955, −5.26259218881425705491593562626, −4.27630864557566518018787456996, −3.39455638567554239278118331441, −1.94363061949545825485439880304, −0.905665913566149566844231977803,
1.57524137386085790187701855969, 2.35336805806330563720610955749, 3.69671675833016336473403765613, 4.52740208341722857855538290370, 5.89977247781588452141840122716, 6.28663596806027908069130909675, 6.94959191496572905790491693111, 8.291088575153106100783681739681, 9.107199685103969193972183334531, 9.356617050526758854595170777932
