| L(s) = 1 | + 4.23·2-s − 3·3-s + 9.94·4-s − 12.7·6-s + 7·7-s + 8.23·8-s + 9·9-s − 41.5·11-s − 29.8·12-s − 88.9·13-s + 29.6·14-s − 44.6·16-s + 120.·17-s + 38.1·18-s − 112.·19-s − 21·21-s − 175.·22-s + 115.·23-s − 24.7·24-s − 376.·26-s − 27·27-s + 69.6·28-s − 144.·29-s − 258.·31-s − 255.·32-s + 124.·33-s + 509.·34-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 0.577·3-s + 1.24·4-s − 0.864·6-s + 0.377·7-s + 0.363·8-s + 0.333·9-s − 1.13·11-s − 0.717·12-s − 1.89·13-s + 0.566·14-s − 0.697·16-s + 1.71·17-s + 0.499·18-s − 1.35·19-s − 0.218·21-s − 1.70·22-s + 1.04·23-s − 0.210·24-s − 2.84·26-s − 0.192·27-s + 0.469·28-s − 0.927·29-s − 1.49·31-s − 1.40·32-s + 0.657·33-s + 2.57·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.23T + 8T^{2} \) |
| 11 | \( 1 + 41.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 258.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 48.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 575.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 529.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.28T + 3.00e5T^{2} \) |
| 71 | \( 1 + 61.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 484.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 878.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 491.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 415.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
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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
−10.35057332215629770313497635479, −9.294390206870001585603507244779, −7.75436216850545228995659275747, −7.12964300790286180955237914841, −5.82302987643275065571186523767, −5.19879521923737861167554864204, −4.55718767714795734318854868501, −3.23501241262839552323797862952, −2.13201285088636293002690334807, 0,
2.13201285088636293002690334807, 3.23501241262839552323797862952, 4.55718767714795734318854868501, 5.19879521923737861167554864204, 5.82302987643275065571186523767, 7.12964300790286180955237914841, 7.75436216850545228995659275747, 9.294390206870001585603507244779, 10.35057332215629770313497635479
