| L(s) = 1 | + 4·2-s − 24.7·3-s + 16·4-s + 36.1·5-s − 99.1·6-s + 64·8-s + 370.·9-s + 144.·10-s + 155.·11-s − 396.·12-s − 1.15e3·13-s − 894.·15-s + 256·16-s − 1.23e3·17-s + 1.48e3·18-s + 280.·19-s + 577.·20-s + 621.·22-s − 3.48e3·23-s − 1.58e3·24-s − 1.82e3·25-s − 4.63e3·26-s − 3.16e3·27-s − 5.65e3·29-s − 3.57e3·30-s + 2.31e3·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.58·3-s + 0.5·4-s + 0.645·5-s − 1.12·6-s + 0.353·8-s + 1.52·9-s + 0.456·10-s + 0.387·11-s − 0.794·12-s − 1.90·13-s − 1.02·15-s + 0.250·16-s − 1.03·17-s + 1.07·18-s + 0.178·19-s + 0.322·20-s + 0.273·22-s − 1.37·23-s − 0.561·24-s − 0.582·25-s − 1.34·26-s − 0.836·27-s − 1.24·29-s − 0.725·30-s + 0.432·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 24.7T + 243T^{2} \) |
| 5 | \( 1 - 36.1T + 3.12e3T^{2} \) |
| 11 | \( 1 - 155.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.23e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 280.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.01e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.08e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.50e4T + 8.58e9T^{2} \) |
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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
−12.22715536997832448567860500035, −11.67304197434026655224988956812, −10.50310156414275704780815847847, −9.585468814806705078868677982426, −7.36920560691245433444420175771, −6.30136893475681043596166787807, −5.40641087764654157480291848065, −4.37618470967002820033851352433, −2.05618776637421083580884366666, 0,
2.05618776637421083580884366666, 4.37618470967002820033851352433, 5.40641087764654157480291848065, 6.30136893475681043596166787807, 7.36920560691245433444420175771, 9.585468814806705078868677982426, 10.50310156414275704780815847847, 11.67304197434026655224988956812, 12.22715536997832448567860500035
