| L(s) = 1 | − 9.64·5-s + 1.12i·7-s + 13.8i·11-s + 8.65·13-s + 12.6·17-s + 27.6i·19-s − 27.5i·23-s + 68.0·25-s − 6.15·29-s − 13.2i·31-s − 10.8i·35-s + 32.9·37-s − 60.2·41-s − 25.3i·43-s + 66.7i·47-s + ⋯ |
| L(s) = 1 | − 1.92·5-s + 0.160i·7-s + 1.25i·11-s + 0.665·13-s + 0.744·17-s + 1.45i·19-s − 1.19i·23-s + 2.72·25-s − 0.212·29-s − 0.427i·31-s − 0.308i·35-s + 0.890·37-s − 1.46·41-s − 0.589i·43-s + 1.42i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3688162327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3688162327\) |
| \(L(2)\) |
|
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 | \( 1 + 9.64T + 25T^{2} \) |
| 7 | \( 1 - 1.12iT - 49T^{2} \) |
| 11 | \( 1 - 13.8iT - 121T^{2} \) |
| 13 | \( 1 - 8.65T + 169T^{2} \) |
| 17 | \( 1 - 12.6T + 289T^{2} \) |
| 19 | \( 1 - 27.6iT - 361T^{2} \) |
| 23 | \( 1 + 27.5iT - 529T^{2} \) |
| 29 | \( 1 + 6.15T + 841T^{2} \) |
| 31 | \( 1 + 13.2iT - 961T^{2} \) |
| 37 | \( 1 - 32.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 60.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 25.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 13.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 83.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 101.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 112. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 25.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 111. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 22.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 71.2T + 9.40e3T^{2} \) |
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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
−9.527922179144407807275671177294, −8.401762258596121067003646413184, −8.034045645579480203715333431556, −7.30452118168607571757197186412, −6.52395106221037614004540215913, −5.35400062105220583916594546953, −4.27523383204402505536275413166, −3.89538749398499803102512326041, −2.80603133206306845605260989378, −1.29196877589769092398566710375,
0.12513621718732863537364936728, 1.08523839958359691368717247596, 3.11704115904753306691675924406, 3.49771842239869160561286101006, 4.45054574017003508442029366755, 5.37262565941003674680711650318, 6.47165709213631010355767550120, 7.34666293656469625143407405244, 7.926946688099894726965808662774, 8.624047262202164156238337869892
