| L(s) = 1 | − 0.123·3-s − 101.·5-s − 122.·7-s − 242.·9-s − 199.·11-s − 69.1·13-s + 12.6·15-s − 1.58e3·17-s + 2.85e3·19-s + 15.1·21-s + 3.76e3·23-s + 7.25e3·25-s + 60.1·27-s + 3.37e3·29-s − 2.94e3·31-s + 24.7·33-s + 1.24e4·35-s + 5.91e3·37-s + 8.56·39-s + 1.26e4·41-s + 2.48e3·43-s + 2.47e4·45-s − 8.07e3·47-s − 1.78e3·49-s + 196.·51-s − 2.46e3·53-s + 2.03e4·55-s + ⋯ |
| L(s) = 1 | − 0.00794·3-s − 1.82·5-s − 0.945·7-s − 0.999·9-s − 0.497·11-s − 0.113·13-s + 0.0144·15-s − 1.32·17-s + 1.81·19-s + 0.00751·21-s + 1.48·23-s + 2.32·25-s + 0.0158·27-s + 0.745·29-s − 0.549·31-s + 0.00395·33-s + 1.72·35-s + 0.710·37-s + 0.000901·39-s + 1.17·41-s + 0.204·43-s + 1.82·45-s − 0.533·47-s − 0.106·49-s + 0.0105·51-s − 0.120·53-s + 0.906·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 0.123T + 243T^{2} \) |
| 5 | \( 1 + 101.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 122.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 199.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 69.1T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.85e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.91e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.26e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.48e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.07e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.46e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.64e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.99e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.67e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.18e5T + 8.58e9T^{2} \) |
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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
−8.815593036671797437025658211559, −7.85328635793597876681661744678, −7.27521499437150857684124068553, −6.39351595068967864130081624897, −5.20481933559523929284965104859, −4.35053039132626313940138609911, −3.19498443583787573241557980661, −2.87505473017886111605563233426, −0.77694010932928882457502615296, 0,
0.77694010932928882457502615296, 2.87505473017886111605563233426, 3.19498443583787573241557980661, 4.35053039132626313940138609911, 5.20481933559523929284965104859, 6.39351595068967864130081624897, 7.27521499437150857684124068553, 7.85328635793597876681661744678, 8.815593036671797437025658211559
