L(s) = 1 | + i·3-s + i·5-s + 4·7-s − 9-s + 2i·13-s − 15-s + 6·17-s − 4i·19-s + 4i·21-s − 25-s − i·27-s − 6i·29-s + 8·31-s + 4i·35-s − 2i·37-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s + 1.51·7-s − 0.333·9-s + 0.554i·13-s − 0.258·15-s + 1.45·17-s − 0.917i·19-s + 0.872i·21-s − 0.200·25-s − 0.192i·27-s − 1.11i·29-s + 1.43·31-s + 0.676i·35-s − 0.328i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.513961194\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.513961194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.421374187747940337034609501809, −7.953902831451915862831395363159, −7.26069398724145297566384399809, −6.26946337792334902790253549379, −5.49899491284154989945564468472, −4.69294558814507026063417205080, −4.18193085068625208485071899012, −3.07209528420726367066002337904, −2.19627369630982448406835930496, −1.01473690259921246217950180308,
0.995658404420836166060860145934, 1.58773439016140608985161131265, 2.73657141623597950703662076789, 3.78401217892843558689289440772, 4.78170539601653697535486997058, 5.40475459459180967659541159039, 6.01307665431196708029310950435, 7.14699917627277091601017031436, 7.83108482472909831459656684776, 8.215407454374943375546659803878