L(s) = 1 | + 1.41i·2-s − 4.30i·3-s − 2.00·4-s + 6.09·6-s + 1.47·7-s − 2.82i·8-s − 9.55·9-s − 6.04i·11-s + 8.61i·12-s + 5.21i·13-s + 2.08i·14-s + 4.00·16-s − 15.7·17-s − 13.5i·18-s − 4.82i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.43i·3-s − 0.500·4-s + 1.01·6-s + 0.210·7-s − 0.353i·8-s − 1.06·9-s − 0.549i·11-s + 0.717i·12-s + 0.401i·13-s + 0.149i·14-s + 0.250·16-s − 0.923·17-s − 0.750i·18-s − 0.254i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1852240447\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1852240447\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (22.8 - 2.58i)T \) |
good | 3 | \( 1 + 4.30iT - 9T^{2} \) |
| 7 | \( 1 - 1.47T + 49T^{2} \) |
| 11 | \( 1 + 6.04iT - 121T^{2} \) |
| 13 | \( 1 - 5.21iT - 169T^{2} \) |
| 17 | \( 1 + 15.7T + 289T^{2} \) |
| 19 | \( 1 + 4.82iT - 361T^{2} \) |
| 29 | \( 1 - 23.4T + 841T^{2} \) |
| 31 | \( 1 + 20.4T + 961T^{2} \) |
| 37 | \( 1 + 15.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 20.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 38.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 13.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 100. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 32.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 15.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 11.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 44.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 154.T + 9.40e3T^{2} \) |
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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
−8.739550602068476317154058993427, −8.210697435627961341842040818679, −7.40130829386831379406847727496, −6.66292437481218921984463214071, −6.12897201948963898269758512842, −5.08394472568685562578549340314, −3.94043089348966173653861240254, −2.50065834808373384964422138713, −1.38681493762216397935370567574, −0.05503391404585677687220681785,
1.84321149831552417579942478106, 3.06774574640196497714099260717, 4.01085966703103517960936511055, 4.66304396935229865805847105401, 5.43164579886655845456120680697, 6.61541270706620488656993403847, 7.933487890650773883691800170824, 8.695104754381657102586912915036, 9.577189982747236482296262057472, 10.01357059574841490644661537573