| L(s) = 1 | + (0.978 + 0.207i)2-s + (1.61 + 0.612i)3-s + (0.913 + 0.406i)4-s + (2.21 − 0.290i)5-s + (1.45 + 0.936i)6-s + (0.132 − 0.229i)7-s + (0.809 + 0.587i)8-s + (2.24 + 1.98i)9-s + (2.22 + 0.177i)10-s + (−3.86 − 0.821i)11-s + (1.23 + 1.21i)12-s + (−3.18 + 0.676i)13-s + (0.177 − 0.197i)14-s + (3.76 + 0.888i)15-s + (0.669 + 0.743i)16-s + (−3.78 − 2.75i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (0.935 + 0.353i)3-s + (0.456 + 0.203i)4-s + (0.991 − 0.129i)5-s + (0.594 + 0.382i)6-s + (0.0501 − 0.0869i)7-s + (0.286 + 0.207i)8-s + (0.749 + 0.661i)9-s + (0.704 + 0.0559i)10-s + (−1.16 − 0.247i)11-s + (0.355 + 0.351i)12-s + (−0.883 + 0.187i)13-s + (0.0474 − 0.0527i)14-s + (0.973 + 0.229i)15-s + (0.167 + 0.185i)16-s + (−0.918 − 0.667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.88169 + 0.693694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.88169 + 0.693694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 3 | \( 1 + (-1.61 - 0.612i)T \) |
| 5 | \( 1 + (-2.21 + 0.290i)T \) |
| good | 7 | \( 1 + (-0.132 + 0.229i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.86 + 0.821i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.18 - 0.676i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.78 + 2.75i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (4.92 + 3.58i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.53 + 2.81i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.628 - 5.98i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.253 + 2.40i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (2.82 - 8.68i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.58 + 0.761i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.506 + 0.876i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0870 + 0.828i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-4.19 + 3.04i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.61 + 0.343i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (9.94 + 2.11i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (1.45 - 13.8i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-10.6 + 7.74i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.32 + 7.15i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.773 + 7.36i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-14.0 + 6.23i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (3.46 + 10.6i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 15.5i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−10.88650378549831937280744618293, −10.36569271039653651410641457701, −9.263371078117971605135988317246, −8.561290652849687809346662270060, −7.38304666829626516124255034127, −6.51649622919209878581146149652, −5.07525264350442479962823745215, −4.59783813536558520958814107638, −2.88686642731267263784845833586, −2.24268632177584747358183835096,
2.00785145382572598697039497361, 2.61893909793276948562199089699, 4.06841717104017128691017791081, 5.27410808436298647257259665217, 6.30009968125615649347151305348, 7.27788190232038312286570614193, 8.232941146973116289479105520631, 9.303510301997148475190253500942, 10.19080942527888231706918402988, 10.84600318292008860181815528166
