| L(s) = 1 | + 9.15·2-s + 51.7·4-s − 3.86·5-s − 226.·7-s + 180.·8-s − 35.3·10-s − 18.3·11-s − 711.·13-s − 2.07e3·14-s − 3.60·16-s − 165.·17-s + 213.·19-s − 199.·20-s − 167.·22-s + 529·23-s − 3.11e3·25-s − 6.50e3·26-s − 1.17e4·28-s − 5.47e3·29-s + 6.07e3·31-s − 5.80e3·32-s − 1.51e3·34-s + 874.·35-s + 1.14e4·37-s + 1.95e3·38-s − 697.·40-s + 1.31e4·41-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.61·4-s − 0.0691·5-s − 1.74·7-s + 0.997·8-s − 0.111·10-s − 0.0456·11-s − 1.16·13-s − 2.82·14-s − 0.00352·16-s − 0.138·17-s + 0.135·19-s − 0.111·20-s − 0.0738·22-s + 0.208·23-s − 0.995·25-s − 1.88·26-s − 2.82·28-s − 1.20·29-s + 1.13·31-s − 1.00·32-s − 0.224·34-s + 0.120·35-s + 1.38·37-s + 0.219·38-s − 0.0689·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 9.15T + 32T^{2} \) |
| 5 | \( 1 + 3.86T + 3.12e3T^{2} \) |
| 7 | \( 1 + 226.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 18.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 711.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 165.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 213.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 5.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.31e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.91e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.17e5T + 8.58e9T^{2} \) |
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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
−11.51653537697617144882442944349, −10.10254363334194766677251637998, −9.317661098367221257190960761298, −7.49316661228946297773329676192, −6.52097934487642554075789882332, −5.72390330825890519459036008403, −4.47237903113161381117725941419, −3.38609960042621845737020439760, −2.47835561951525875233903615179, 0,
2.47835561951525875233903615179, 3.38609960042621845737020439760, 4.47237903113161381117725941419, 5.72390330825890519459036008403, 6.52097934487642554075789882332, 7.49316661228946297773329676192, 9.317661098367221257190960761298, 10.10254363334194766677251637998, 11.51653537697617144882442944349
