L(s) = 1 | − 3.27i·2-s − 3i·3-s − 2.75·4-s − 9.83·6-s + 33.3i·7-s − 17.2i·8-s − 9·9-s − 11·11-s + 8.26i·12-s − 24.2i·13-s + 109.·14-s − 78.4·16-s − 69.7i·17-s + 29.5i·18-s + 125.·19-s + ⋯ |
L(s) = 1 | − 1.15i·2-s − 0.577i·3-s − 0.344·4-s − 0.669·6-s + 1.79i·7-s − 0.760i·8-s − 0.333·9-s − 0.301·11-s + 0.198i·12-s − 0.518i·13-s + 2.08·14-s − 1.22·16-s − 0.994i·17-s + 0.386i·18-s + 1.51·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.933599418\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.933599418\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 + 3.27iT - 8T^{2} \) |
| 7 | \( 1 - 33.3iT - 343T^{2} \) |
| 13 | \( 1 + 24.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 69.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 238.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 166. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 585. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 40.0iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 391.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 858. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 877. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 9.99e2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3iT - 9.12e5T^{2} \) |
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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
−9.442900178338559193611062746882, −8.976143019888749722775010609421, −7.81110714958061512935636206928, −6.95113288471521937027187304598, −5.72714227480464831253349451474, −5.15128348543503557424487829908, −3.38742940117793666044599525656, −2.69714838672851145114810526859, −1.87262720620055642839343324090, −0.56850466959310372496269569540,
1.11215058160818887827375167001, 2.94063814150439042296339638387, 4.23002191741753153766090178095, 4.79765703751663101373928634863, 6.03744910250254503021457198689, 6.75648368076215764087066279563, 7.61665749445122091926666594540, 8.159913908647039216635539652601, 9.276007606464331604213172606880, 10.22606661904706443009484781176