| L(s) = 1 | − 5.44·2-s − 2.33·4-s − 36·5-s + 187.·8-s + 196.·10-s − 184.·11-s + 147.·13-s − 943.·16-s − 1.96e3·17-s + 1.89e3·19-s + 84.1·20-s + 1.00e3·22-s − 136.·23-s − 1.82e3·25-s − 805.·26-s + 1.25e3·29-s + 8.96e3·31-s − 844.·32-s + 1.07e4·34-s + 1.28e4·37-s − 1.03e4·38-s − 6.73e3·40-s − 8.97e3·41-s + 1.35e4·43-s + 431.·44-s + 746.·46-s + 2.00e4·47-s + ⋯ |
| L(s) = 1 | − 0.962·2-s − 0.0730·4-s − 0.643·5-s + 1.03·8-s + 0.620·10-s − 0.459·11-s + 0.242·13-s − 0.921·16-s − 1.65·17-s + 1.20·19-s + 0.0470·20-s + 0.442·22-s − 0.0539·23-s − 0.585·25-s − 0.233·26-s + 0.278·29-s + 1.67·31-s − 0.145·32-s + 1.59·34-s + 1.54·37-s − 1.15·38-s − 0.665·40-s − 0.833·41-s + 1.11·43-s + 0.0335·44-s + 0.0519·46-s + 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.44T + 32T^{2} \) |
| 5 | \( 1 + 36T + 3.12e3T^{2} \) |
| 11 | \( 1 + 184.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.96e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.89e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 136.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.96e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.33e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.86e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.18e5T + 8.58e9T^{2} \) |
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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
−9.712709122393030863485025696978, −8.972605405864452239929932479973, −8.061160512203894297825318278694, −7.49154779494772117120694906896, −6.31474891901061091744784120983, −4.87383340436066500426176727243, −4.04871306593225445512755470686, −2.54203190746228453997094126595, −1.04691134463053307894317112920, 0,
1.04691134463053307894317112920, 2.54203190746228453997094126595, 4.04871306593225445512755470686, 4.87383340436066500426176727243, 6.31474891901061091744784120983, 7.49154779494772117120694906896, 8.061160512203894297825318278694, 8.972605405864452239929932479973, 9.712709122393030863485025696978
