| L(s) = 1 | − 5.29·5-s − 7.48i·7-s − 5.65i·11-s + 4·13-s − 21.1·17-s + 29.9i·19-s + 22.6i·23-s + 3.00·25-s + 5.29·29-s + 22.4i·31-s + 39.5i·35-s + 28·37-s − 63.4·41-s + 29.9i·43-s − 67.8i·47-s + ⋯ |
| L(s) = 1 | − 1.05·5-s − 1.06i·7-s − 0.514i·11-s + 0.307·13-s − 1.24·17-s + 1.57i·19-s + 0.983i·23-s + 0.120·25-s + 0.182·29-s + 0.724i·31-s + 1.13i·35-s + 0.756·37-s − 1.54·41-s + 0.696i·43-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.179682872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179682872\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.29T + 25T^{2} \) |
| 7 | \( 1 + 7.48iT - 49T^{2} \) |
| 11 | \( 1 + 5.65iT - 121T^{2} \) |
| 13 | \( 1 - 4T + 169T^{2} \) |
| 17 | \( 1 + 21.1T + 289T^{2} \) |
| 19 | \( 1 - 29.9iT - 361T^{2} \) |
| 23 | \( 1 - 22.6iT - 529T^{2} \) |
| 29 | \( 1 - 5.29T + 841T^{2} \) |
| 31 | \( 1 - 22.4iT - 961T^{2} \) |
| 37 | \( 1 - 28T + 1.36e3T^{2} \) |
| 41 | \( 1 + 63.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 29.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 47.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 76T + 3.72e3T^{2} \) |
| 67 | \( 1 + 59.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 26T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 42.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 18T + 9.40e3T^{2} \) |
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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
−8.493406560043584215116903247169, −8.171544364037933207790225626074, −7.28305839305488244644933691799, −6.69546405048809507884362334056, −5.70104965447059024730672110360, −4.64526567078014076219718317328, −3.79397364652089705880713046718, −3.42675539172334277384849347468, −1.81535743820746506401413712009, −0.61057517526846142974347719454,
0.49842633673471309395348560732, 2.17554050504794753722698984454, 2.87297003141255478794399002453, 4.15539205952873212124617729127, 4.64511449190272665486274384460, 5.67396327195047895875731309670, 6.63671538824298935293623898936, 7.22034173717963695591212003542, 8.216484047191973413879996233260, 8.768377174416188813138897231699
