| L(s) = 1 | − 4.92·2-s + 16.2·4-s − 34.7·7-s − 40.7·8-s + 12.4·11-s − 37.1·13-s + 171.·14-s + 70.7·16-s + 102.·17-s + 63.7·19-s − 61.3·22-s + 100.·23-s + 182.·26-s − 566.·28-s − 138.·29-s + 24.8·31-s − 22.4·32-s − 505.·34-s − 186.·37-s − 314.·38-s + 255.·41-s + 108.·43-s + 202.·44-s − 495.·46-s − 223.·47-s + 866.·49-s − 604.·52-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.03·4-s − 1.87·7-s − 1.80·8-s + 0.341·11-s − 0.791·13-s + 3.27·14-s + 1.10·16-s + 1.46·17-s + 0.770·19-s − 0.594·22-s + 0.911·23-s + 1.37·26-s − 3.82·28-s − 0.889·29-s + 0.144·31-s − 0.123·32-s − 2.55·34-s − 0.829·37-s − 1.34·38-s + 0.971·41-s + 0.384·43-s + 0.694·44-s − 1.58·46-s − 0.694·47-s + 2.52·49-s − 1.61·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.92T + 8T^{2} \) |
| 7 | \( 1 + 34.7T + 343T^{2} \) |
| 11 | \( 1 - 12.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 63.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 24.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 186.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 255.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 40.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 131.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 336.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 17.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 992.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 769.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 252.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 234.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 405.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−9.472930750848402927878217508143, −9.263122594057430024645220055202, −7.947286809549849393980009669130, −7.19778449463729888397509458726, −6.54367253416161515133678399812, −5.48255811944818135030179827930, −3.52283345375779452751698960259, −2.65779152409578097574074449784, −1.09222483462991891044130810889, 0,
1.09222483462991891044130810889, 2.65779152409578097574074449784, 3.52283345375779452751698960259, 5.48255811944818135030179827930, 6.54367253416161515133678399812, 7.19778449463729888397509458726, 7.947286809549849393980009669130, 9.263122594057430024645220055202, 9.472930750848402927878217508143
