| L(s) = 1 | + 19.4·2-s − 27·3-s + 251.·4-s + 181.·5-s − 525.·6-s − 198.·7-s + 2.40e3·8-s + 729·9-s + 3.53e3·10-s − 6.78e3·12-s − 4.87e3·13-s − 3.86e3·14-s − 4.89e3·15-s + 1.46e4·16-s − 1.84e4·17-s + 1.41e4·18-s − 5.79e4·19-s + 4.55e4·20-s + 5.36e3·21-s − 4.38e4·23-s − 6.48e4·24-s − 4.52e4·25-s − 9.48e4·26-s − 1.96e4·27-s − 4.99e4·28-s + 2.10e5·29-s − 9.54e4·30-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.96·4-s + 0.649·5-s − 0.993·6-s − 0.218·7-s + 1.65·8-s + 0.333·9-s + 1.11·10-s − 1.13·12-s − 0.615·13-s − 0.376·14-s − 0.374·15-s + 0.891·16-s − 0.911·17-s + 0.573·18-s − 1.93·19-s + 1.27·20-s + 0.126·21-s − 0.751·23-s − 0.957·24-s − 0.578·25-s − 1.05·26-s − 0.192·27-s − 0.429·28-s + 1.60·29-s − 0.645·30-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 19.4T + 128T^{2} \) |
| 5 | \( 1 - 181.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 198.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 4.87e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.84e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.10e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.58e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.55e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.93e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.54e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.92e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.90e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.79e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.02e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.42e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.68e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.28e6T + 8.07e13T^{2} \) |
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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
−10.21890593100036072367108826252, −8.903204631391315775917451367247, −7.37516314193634628578897960018, −6.25260966171124212129413004673, −6.02467699100200785431925595902, −4.71035448271786269984009678293, −4.18181728626594880505603948454, −2.68785782212124811854383419133, −1.87904664284470825650501162934, 0,
1.87904664284470825650501162934, 2.68785782212124811854383419133, 4.18181728626594880505603948454, 4.71035448271786269984009678293, 6.02467699100200785431925595902, 6.25260966171124212129413004673, 7.37516314193634628578897960018, 8.903204631391315775917451367247, 10.21890593100036072367108826252
