L(s) = 1 | + 2.04·2-s − 27.8·4-s − 42.3·5-s + 191.·7-s − 122.·8-s − 86.4·10-s + 655.·11-s − 932.·13-s + 391.·14-s + 641.·16-s + 344.·17-s − 548.·19-s + 1.17e3·20-s + 1.33e3·22-s + 529·23-s − 1.33e3·25-s − 1.90e3·26-s − 5.33e3·28-s − 4.39e3·29-s − 4.43e3·31-s + 5.21e3·32-s + 702.·34-s − 8.11e3·35-s − 1.14e4·37-s − 1.11e3·38-s + 5.17e3·40-s + 6.66e3·41-s + ⋯ |
L(s) = 1 | + 0.360·2-s − 0.869·4-s − 0.757·5-s + 1.47·7-s − 0.674·8-s − 0.273·10-s + 1.63·11-s − 1.53·13-s + 0.533·14-s + 0.626·16-s + 0.288·17-s − 0.348·19-s + 0.659·20-s + 0.589·22-s + 0.208·23-s − 0.425·25-s − 0.552·26-s − 1.28·28-s − 0.971·29-s − 0.828·31-s + 0.900·32-s + 0.104·34-s − 1.12·35-s − 1.37·37-s − 0.125·38-s + 0.511·40-s + 0.618·41-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 | \( 1 - 2.04T + 32T^{2} \) |
| 5 | \( 1 + 42.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 191.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 655.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 932.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 344.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 548.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 4.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.51e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.21e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.80e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.37e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.22e4T + 8.58e9T^{2} \) |
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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
−11.46339461494773579956165603883, −9.942262375578843200778828651903, −8.937244627866826920026453541290, −8.069789871867066558931947782610, −7.05371593967462817077313348123, −5.37835105955995206111827621247, −4.52245238586583310700619531510, −3.65337288200133782614945570384, −1.63564036329848118343167978314, 0,
1.63564036329848118343167978314, 3.65337288200133782614945570384, 4.52245238586583310700619531510, 5.37835105955995206111827621247, 7.05371593967462817077313348123, 8.069789871867066558931947782610, 8.937244627866826920026453541290, 9.942262375578843200778828651903, 11.46339461494773579956165603883