| L(s) = 1 | − 4·2-s + 16·4-s + 141.·7-s − 64·8-s + 113.·11-s + 61.7·13-s − 566.·14-s + 256·16-s − 1.67e3·17-s − 662.·19-s − 453.·22-s − 86.4·23-s − 246.·26-s + 2.26e3·28-s − 3.23e3·29-s − 3.81e3·31-s − 1.02e3·32-s + 6.68e3·34-s + 1.02e4·37-s + 2.64e3·38-s − 1.34e4·41-s − 4.69e3·43-s + 1.81e3·44-s + 345.·46-s + 1.52e4·47-s + 3.25e3·49-s + 987.·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.09·7-s − 0.353·8-s + 0.282·11-s + 0.101·13-s − 0.772·14-s + 0.250·16-s − 1.40·17-s − 0.420·19-s − 0.199·22-s − 0.0340·23-s − 0.0716·26-s + 0.546·28-s − 0.713·29-s − 0.712·31-s − 0.176·32-s + 0.991·34-s + 1.23·37-s + 0.297·38-s − 1.25·41-s − 0.387·43-s + 0.141·44-s + 0.0241·46-s + 1.00·47-s + 0.193·49-s + 0.0506·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 113.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 61.7T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.67e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 662.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 86.4T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.52e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 498.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.35e4T + 8.58e9T^{2} \) |
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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
−9.730010082989014656764443620382, −8.849881017281529198956222890742, −8.163463915269753914499360921937, −7.21184072523691576174814792880, −6.25662360636231175816067149148, −5.03518499464291991208628079695, −3.96062384156805589814314549504, −2.36100319155919208136439229256, −1.43152765481458509291591349738, 0,
1.43152765481458509291591349738, 2.36100319155919208136439229256, 3.96062384156805589814314549504, 5.03518499464291991208628079695, 6.25662360636231175816067149148, 7.21184072523691576174814792880, 8.163463915269753914499360921937, 8.849881017281529198956222890742, 9.730010082989014656764443620382
