| L(s) = 1 | − 3.31·3-s + 5-s − 0.477·7-s + 7.95·9-s + 4.57·11-s + 2.86·13-s − 3.31·15-s + 7.02·17-s − 5.71·19-s + 1.58·21-s − 1.60·23-s + 25-s − 16.4·27-s − 8.07·29-s − 11.0·31-s − 15.1·33-s − 0.477·35-s − 2.92·37-s − 9.49·39-s − 5.80·41-s − 7.00·43-s + 7.95·45-s + 12.4·47-s − 6.77·49-s − 23.2·51-s − 0.250·53-s + 4.57·55-s + ⋯ |
| L(s) = 1 | − 1.91·3-s + 0.447·5-s − 0.180·7-s + 2.65·9-s + 1.37·11-s + 0.795·13-s − 0.854·15-s + 1.70·17-s − 1.31·19-s + 0.344·21-s − 0.334·23-s + 0.200·25-s − 3.15·27-s − 1.49·29-s − 1.97·31-s − 2.63·33-s − 0.0807·35-s − 0.481·37-s − 1.52·39-s − 0.905·41-s − 1.06·43-s + 1.18·45-s + 1.80·47-s − 0.967·49-s − 3.25·51-s − 0.0344·53-s + 0.616·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 3.31T + 3T^{2} \) |
| 7 | \( 1 + 0.477T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 - 7.02T + 17T^{2} \) |
| 19 | \( 1 + 5.71T + 19T^{2} \) |
| 23 | \( 1 + 1.60T + 23T^{2} \) |
| 29 | \( 1 + 8.07T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 + 5.80T + 41T^{2} \) |
| 43 | \( 1 + 7.00T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 0.250T + 53T^{2} \) |
| 59 | \( 1 - 2.86T + 59T^{2} \) |
| 61 | \( 1 - 2.93T + 61T^{2} \) |
| 67 | \( 1 - 0.359T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 6.69T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 4.08T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.37723336762683207814474082624, −6.76437954445814204070107002195, −6.22999373044116645796571433479, −5.58777664892392364916091588867, −5.21076739593661041090491637929, −3.92248935156280505809378007264, −3.75209336387424291607077232325, −1.80201326032035768013702380445, −1.25616340787946005613562024557, 0,
1.25616340787946005613562024557, 1.80201326032035768013702380445, 3.75209336387424291607077232325, 3.92248935156280505809378007264, 5.21076739593661041090491637929, 5.58777664892392364916091588867, 6.22999373044116645796571433479, 6.76437954445814204070107002195, 7.37723336762683207814474082624
