| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 4.47i·7-s − i·8-s − 9-s − 0.763·11-s + i·12-s + 4.47i·13-s − 4.47·14-s + 16-s − 4i·17-s − i·18-s + 7.70·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.69i·7-s − 0.353i·8-s − 0.333·9-s − 0.230·11-s + 0.288i·12-s + 1.24i·13-s − 1.19·14-s + 0.250·16-s − 0.970i·17-s − 0.235i·18-s + 1.76·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.270313743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.270313743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.76iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9.23iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 0.763iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 3.70iT - 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 4.47iT - 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 8.76iT - 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 8.47iT - 97T^{2} \) |
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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
−8.955496821973655785454602685812, −8.019679683381813015569892431716, −7.48965004749964806468762276689, −6.55722470971877508319688670232, −6.07854451236920206309421897485, −5.19355362904073857097440551305, −4.71224367663846975192229898838, −3.22453283988349156283894718667, −2.50609533582466726981959431491, −1.34556723864111147544652965985,
0.41519552243997417574454857071, 1.39295367290422017592336072828, 2.85430583365018766061487902148, 3.64094891403313534186395946773, 4.10099183399613908405686510051, 5.15460250926479871648070220625, 5.70778765286829404271475043635, 6.96348350719852997241722343081, 7.69082249887034208491230497172, 8.221698363283947130702600940561
