| L(s) = 1 | + 9.67·2-s + 27·3-s − 34.3·4-s + 124.·5-s + 261.·6-s + 573.·7-s − 1.57e3·8-s + 729·9-s + 1.20e3·10-s − 928.·12-s − 3.16e3·13-s + 5.54e3·14-s + 3.36e3·15-s − 1.07e4·16-s + 1.85e4·17-s + 7.05e3·18-s − 2.44e3·19-s − 4.29e3·20-s + 1.54e4·21-s + 5.35e4·23-s − 4.24e4·24-s − 6.25e4·25-s − 3.06e4·26-s + 1.96e4·27-s − 1.97e4·28-s + 4.99e4·29-s + 3.25e4·30-s + ⋯ |
| L(s) = 1 | + 0.855·2-s + 0.577·3-s − 0.268·4-s + 0.446·5-s + 0.493·6-s + 0.631·7-s − 1.08·8-s + 0.333·9-s + 0.381·10-s − 0.155·12-s − 0.399·13-s + 0.540·14-s + 0.257·15-s − 0.659·16-s + 0.916·17-s + 0.285·18-s − 0.0818·19-s − 0.119·20-s + 0.364·21-s + 0.918·23-s − 0.626·24-s − 0.800·25-s − 0.341·26-s + 0.192·27-s − 0.169·28-s + 0.380·29-s + 0.220·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.462013586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.462013586\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 9.67T + 128T^{2} \) |
| 5 | \( 1 - 124.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 573.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 3.16e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.85e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.44e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.99e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.08e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.76e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.25e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.25e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.06e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 8.88e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.33e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.68e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.89e6T + 8.07e13T^{2} \) |
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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
−10.06050827934634911068130005339, −9.342180044702935425338123127186, −8.397195787267468582898288844685, −7.46599987746958997610018703716, −6.14323189190207302311939972209, −5.21593313594184899251210699676, −4.36691705813158451048485005557, −3.28531037594794619580817099680, −2.25626974846670760185498950891, −0.871463512241752184363706745975,
0.871463512241752184363706745975, 2.25626974846670760185498950891, 3.28531037594794619580817099680, 4.36691705813158451048485005557, 5.21593313594184899251210699676, 6.14323189190207302311939972209, 7.46599987746958997610018703716, 8.397195787267468582898288844685, 9.342180044702935425338123127186, 10.06050827934634911068130005339
