| L(s) = 1 | + (−0.743 − 0.669i)2-s + (−0.417 + 1.68i)3-s + (0.104 + 0.994i)4-s + (0.153 − 2.23i)5-s + (1.43 − 0.969i)6-s + (−1.71 + 0.991i)7-s + (0.587 − 0.809i)8-s + (−2.65 − 1.40i)9-s + (−1.60 + 1.55i)10-s + (0.327 − 0.364i)11-s + (−1.71 − 0.239i)12-s + (−2.77 + 2.49i)13-s + (1.93 + 0.412i)14-s + (3.68 + 1.19i)15-s + (−0.978 + 0.207i)16-s + (−0.627 + 0.864i)17-s + ⋯ |
| L(s) = 1 | + (−0.525 − 0.473i)2-s + (−0.241 + 0.970i)3-s + (0.0522 + 0.497i)4-s + (0.0685 − 0.997i)5-s + (0.585 − 0.395i)6-s + (−0.648 + 0.374i)7-s + (0.207 − 0.286i)8-s + (−0.883 − 0.468i)9-s + (−0.508 + 0.491i)10-s + (0.0988 − 0.109i)11-s + (−0.495 − 0.0692i)12-s + (−0.769 + 0.692i)13-s + (0.518 + 0.110i)14-s + (0.951 + 0.307i)15-s + (−0.244 + 0.0519i)16-s + (−0.152 + 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00417952 - 0.0612198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00417952 - 0.0612198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.743 + 0.669i)T \) |
| 3 | \( 1 + (0.417 - 1.68i)T \) |
| 5 | \( 1 + (-0.153 + 2.23i)T \) |
| good | 7 | \( 1 + (1.71 - 0.991i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.327 + 0.364i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.77 - 2.49i)T + (1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.627 - 0.864i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (5.09 + 3.69i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.301 - 1.41i)T + (-21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (4.50 + 2.00i)T + (19.4 + 21.5i)T^{2} \) |
| 31 | \( 1 + (-3.56 + 1.58i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (10.7 + 3.48i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (6.62 + 7.35i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (0.694 - 0.400i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.225 - 0.505i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-1.66 - 2.29i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-5.77 - 6.41i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.0777 - 0.0863i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.80 - 10.8i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-6.65 + 4.83i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (5.73 - 1.86i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.17 - 0.969i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.518 - 0.0545i)T + (81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (2.68 + 8.25i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (6.77 - 15.2i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−10.49772266512905508675447961747, −9.685562827732558513878388181844, −9.020849979349440718306262351433, −8.484408048623436923514451913245, −6.95010486963648866281148653219, −5.74596142098773648239369316431, −4.67851266355331645901378507721, −3.75535485147091271837629013426, −2.25124774028534053868151090855, −0.04311530833203755341337712063,
2.01543771585769878420391277382, 3.33541200641603041544472436685, 5.21014205938247004592861722254, 6.41305664177394856112260395684, 6.79793161987376007300364569222, 7.71012579555056451071005001845, 8.533902536670375036007051657930, 9.942249281819654384958244991958, 10.43035603655925919012883764000, 11.42057532834838360968386984940
