| L(s) = 1 | + 9.77·3-s − 24.1·5-s − 64.5·7-s − 147.·9-s + 58.5·11-s + 875.·13-s − 235.·15-s + 2.05e3·17-s + 361·19-s − 630.·21-s − 4.90e3·23-s − 2.54e3·25-s − 3.81e3·27-s + 1.70e3·29-s − 3.65e3·31-s + 572.·33-s + 1.55e3·35-s − 2.20e3·37-s + 8.55e3·39-s − 1.22e4·41-s − 1.09e4·43-s + 3.56e3·45-s − 1.21e4·47-s − 1.26e4·49-s + 2.00e4·51-s − 9.12e3·53-s − 1.41e3·55-s + ⋯ |
| L(s) = 1 | + 0.626·3-s − 0.431·5-s − 0.498·7-s − 0.607·9-s + 0.146·11-s + 1.43·13-s − 0.270·15-s + 1.72·17-s + 0.229·19-s − 0.312·21-s − 1.93·23-s − 0.813·25-s − 1.00·27-s + 0.376·29-s − 0.682·31-s + 0.0915·33-s + 0.215·35-s − 0.264·37-s + 0.900·39-s − 1.14·41-s − 0.900·43-s + 0.262·45-s − 0.799·47-s − 0.751·49-s + 1.07·51-s − 0.446·53-s − 0.0630·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 3 | \( 1 - 9.77T + 243T^{2} \) |
| 5 | \( 1 + 24.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 64.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 58.5T + 1.61e5T^{2} \) |
| 13 | \( 1 - 875.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.05e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.20e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.05e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.16e5T + 8.58e9T^{2} \) |
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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
−10.26851052082049276296623717865, −9.455375377454774988993509910326, −8.295993582423570803533357634303, −7.88259485605608500798783755705, −6.39638176410047841959728458187, −5.53560707873664860120887867208, −3.77473152760989541710456434767, −3.26438969489095388858533831138, −1.62874941465080835005106741390, 0,
1.62874941465080835005106741390, 3.26438969489095388858533831138, 3.77473152760989541710456434767, 5.53560707873664860120887867208, 6.39638176410047841959728458187, 7.88259485605608500798783755705, 8.295993582423570803533357634303, 9.455375377454774988993509910326, 10.26851052082049276296623717865
