| L(s) = 1 | + 13.9·5-s − 12.9·7-s + 49.9·11-s + 32.9·13-s + 17·17-s − 54.8·19-s − 82.0·23-s + 70.1·25-s − 289.·29-s − 232.·31-s − 181.·35-s − 227.·37-s − 437.·41-s + 158.·43-s + 159.·47-s − 174.·49-s − 376.·53-s + 698.·55-s − 185.·59-s + 861.·61-s + 460.·65-s + 178.·67-s − 1.16e3·71-s + 383.·73-s − 648.·77-s − 254·79-s + 447.·83-s + ⋯ |
| L(s) = 1 | + 1.24·5-s − 0.700·7-s + 1.36·11-s + 0.702·13-s + 0.242·17-s − 0.662·19-s − 0.743·23-s + 0.561·25-s − 1.85·29-s − 1.34·31-s − 0.875·35-s − 1.01·37-s − 1.66·41-s + 0.563·43-s + 0.493·47-s − 0.509·49-s − 0.974·53-s + 1.71·55-s − 0.408·59-s + 1.80·61-s + 0.878·65-s + 0.325·67-s − 1.94·71-s + 0.614·73-s − 0.959·77-s − 0.361·79-s + 0.591·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 5 | \( 1 - 13.9T + 125T^{2} \) |
| 7 | \( 1 + 12.9T + 343T^{2} \) |
| 11 | \( 1 - 49.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 54.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 289.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 232.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 437.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 158.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 159.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 376.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 185.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 861.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 178.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 383.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 254T + 4.93e5T^{2} \) |
| 83 | \( 1 - 447.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 291.T + 9.12e5T^{2} \) |
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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
−8.422024823698766335151986103207, −7.23949633469583897286916265738, −6.50099771903652399390639490435, −5.97982973241466508973774759988, −5.29799673283590802120098783548, −3.95667119618461294722235587627, −3.43409797141716385570576574969, −2.03194944621552592447389685025, −1.49271219230403387270106696696, 0,
1.49271219230403387270106696696, 2.03194944621552592447389685025, 3.43409797141716385570576574969, 3.95667119618461294722235587627, 5.29799673283590802120098783548, 5.97982973241466508973774759988, 6.50099771903652399390639490435, 7.23949633469583897286916265738, 8.422024823698766335151986103207
