L(s) = 1 | + i·3-s − 2.82i·5-s − 4.24·7-s − 9-s + 4i·11-s + 4.24i·13-s + 2.82·15-s + 6·17-s − 2i·19-s − 4.24i·21-s − 2.82·23-s − 3.00·25-s − i·27-s − 5.65i·29-s − 4.24·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.26i·5-s − 1.60·7-s − 0.333·9-s + 1.20i·11-s + 1.17i·13-s + 0.730·15-s + 1.45·17-s − 0.458i·19-s − 0.925i·21-s − 0.589·23-s − 0.600·25-s − 0.192i·27-s − 1.05i·29-s − 0.762·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8250779426\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8250779426\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 4.24iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.909604897159085083536804576112, −7.73217622467130100534050076196, −7.07111757244002012156878556294, −6.07903315075149591637553731397, −5.47801207375033429254323448035, −4.41567189804139057758538742271, −4.04503365591092542172993508321, −2.93973703435870774263508461402, −1.75732365069162476491042768141, −0.29742440522921431267645129678,
1.02813935537344075623608082719, 2.75225514996068289314441612414, 3.15363868999166127924238899229, 3.69256539522853874940136348998, 5.54993374771011092510442369234, 5.94842104612876910506602151704, 6.58237109716037419919532310177, 7.36572444782589162722586947326, 7.949268831110242663252386591884, 8.831520537524956205837633397749