| L(s) = 1 | − 3i·3-s − 2i·5-s − 10·7-s − 9·9-s − 10i·11-s − 6·15-s + 30i·21-s + 21·25-s + 27i·27-s + 50i·29-s − 38·31-s − 30·33-s + 20i·35-s + 18i·45-s + 51·49-s + ⋯ |
| L(s) = 1 | − i·3-s − 0.400i·5-s − 1.42·7-s − 9-s − 0.909i·11-s − 0.400·15-s + 1.42i·21-s + 0.839·25-s + i·27-s + 1.72i·29-s − 1.22·31-s − 0.909·33-s + 0.571i·35-s + 0.400i·45-s + 1.04·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1373585749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1373585749\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 + 2iT - 25T^{2} \) |
| 7 | \( 1 + 10T + 49T^{2} \) |
| 11 | \( 1 + 10iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 50iT - 841T^{2} \) |
| 31 | \( 1 + 38T + 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 94iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 10iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58T + 6.24e3T^{2} \) |
| 83 | \( 1 + 134iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 190T + 9.40e3T^{2} \) |
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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
−10.43035248557573640235050703738, −9.113794001124889926639126798960, −8.831620550277877896122846696761, −7.63398212579587543106380995877, −6.80503639046343421827570143715, −6.09020641133191062633505379240, −5.22398655645033158717428551191, −3.56916677160683329053840820316, −2.78468430053239602587745237198, −1.20395447230219946815278538827,
0.04994528787496934983933144159, 2.44160318751001803912962730972, 3.41780303743324726664567926414, 4.27028590879526056818359667723, 5.41976122649572809557997901164, 6.37148167477301171479241778535, 7.15742536086699550173109596587, 8.364568472834463303685792094815, 9.464146010356835504146729453456, 9.767007081464545290530271006547
