| L(s) = 1 | − 2.23i·2-s + 2.99·4-s − 11.1i·5-s − 24.5i·8-s − 25.0·10-s − 31.0·16-s + 138. i·17-s + 164·19-s − 33.5i·20-s + 98.3i·23-s − 125.·25-s − 232·31-s − 127. i·32-s + 310.·34-s − 366. i·38-s + ⋯ |
| L(s) = 1 | − 0.790i·2-s + 0.374·4-s − 0.999i·5-s − 1.08i·8-s − 0.790·10-s − 0.484·16-s + 1.97i·17-s + 1.98·19-s − 0.374i·20-s + 0.891i·23-s − 1.00·25-s − 1.34·31-s − 0.704i·32-s + 1.56·34-s − 1.56i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.05189 - 1.05189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05189 - 1.05189i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 11.1iT \) |
| good | 2 | \( 1 + 2.23iT - 8T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 138. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 164T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 232T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 545. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 621. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 358T + 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 304T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−15.22154044912984405789724003881, −13.51374791651967017059918599848, −12.51570167502749234044992463822, −11.63856618079614887931866362390, −10.34108650582523083908659026227, −9.134228034675816325519154653218, −7.55886491435439315799155592198, −5.68419233400361906279815777704, −3.71445653822306625951668107148, −1.48944059142701723386272575823,
2.84799502217423045744567467775, 5.39726167887966620649887535723, 6.87583696584929135726137543029, 7.64268166337462406338082967422, 9.495787169864354627196298885071, 10.99865052595189472039068516768, 11.87494613724665095041654224346, 13.84699594673982514302232874476, 14.53397201448028164417709467814, 15.74452160286446100211032089098
