| L(s) = 1 | + 6.31·2-s + 7.83·4-s − 40.1·5-s + 201.·7-s − 152.·8-s − 253.·10-s − 542.·11-s + 734.·13-s + 1.26e3·14-s − 1.21e3·16-s − 1.64e3·17-s − 950.·19-s − 314.·20-s − 3.42e3·22-s − 529·23-s − 1.51e3·25-s + 4.63e3·26-s + 1.57e3·28-s − 466.·29-s + 4.35e3·31-s − 2.77e3·32-s − 1.03e4·34-s − 8.07e3·35-s − 1.52e4·37-s − 6.00e3·38-s + 6.12e3·40-s − 1.04e4·41-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.244·4-s − 0.718·5-s + 1.55·7-s − 0.842·8-s − 0.801·10-s − 1.35·11-s + 1.20·13-s + 1.73·14-s − 1.18·16-s − 1.37·17-s − 0.604·19-s − 0.175·20-s − 1.50·22-s − 0.208·23-s − 0.484·25-s + 1.34·26-s + 0.380·28-s − 0.103·29-s + 0.813·31-s − 0.479·32-s − 1.53·34-s − 1.11·35-s − 1.83·37-s − 0.674·38-s + 0.604·40-s − 0.969·41-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 | \( 1 - 6.31T + 32T^{2} \) |
| 5 | \( 1 + 40.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 201.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 542.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 734.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.64e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 950.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 466.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.52e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.45e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.53e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.28e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.53e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.19e4T + 8.58e9T^{2} \) |
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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
−11.35371383126810622829228190570, −10.46069985983734100799846635344, −8.542494556003658350347392884125, −8.225774720814137527887070028093, −6.70438690206621836621015737367, −5.33479458290195777585773706871, −4.61992873826408720519547451538, −3.60830124034372083740272223709, −2.04400194219626236308487313064, 0,
2.04400194219626236308487313064, 3.60830124034372083740272223709, 4.61992873826408720519547451538, 5.33479458290195777585773706871, 6.70438690206621836621015737367, 8.225774720814137527887070028093, 8.542494556003658350347392884125, 10.46069985983734100799846635344, 11.35371383126810622829228190570
