| L(s) = 1 | − 10.6·3-s − 123.·7-s − 129.·9-s − 121·11-s − 917.·13-s − 223.·17-s + 1.56e3·19-s + 1.31e3·21-s + 1.92e3·23-s + 3.97e3·27-s + 3.73e3·29-s + 8.09e3·31-s + 1.29e3·33-s + 1.75e3·37-s + 9.80e3·39-s − 4.03e3·41-s + 6.22e3·43-s − 3.50e3·47-s − 1.62e3·49-s + 2.38e3·51-s + 4.87e3·53-s − 1.66e4·57-s + 2.18e3·59-s − 1.05e4·61-s + 1.58e4·63-s − 2.15e4·67-s − 2.05e4·69-s + ⋯ |
| L(s) = 1 | − 0.684·3-s − 0.950·7-s − 0.530·9-s − 0.301·11-s − 1.50·13-s − 0.187·17-s + 0.993·19-s + 0.650·21-s + 0.758·23-s + 1.04·27-s + 0.823·29-s + 1.51·31-s + 0.206·33-s + 0.211·37-s + 1.03·39-s − 0.374·41-s + 0.513·43-s − 0.231·47-s − 0.0967·49-s + 0.128·51-s + 0.238·53-s − 0.680·57-s + 0.0815·59-s − 0.363·61-s + 0.504·63-s − 0.585·67-s − 0.519·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 + 10.6T + 243T^{2} \) |
| 7 | \( 1 + 123.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 917.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 223.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.92e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.03e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.22e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.50e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.87e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.18e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.15e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.45e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.92e4T + 8.58e9T^{2} \) |
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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
−8.812633765976544954885098670577, −7.77979793022761275258522686715, −6.90647216259293069350844450107, −6.21597557358647333645832838897, −5.26560035815040153949942892364, −4.63022075865145605392796774799, −3.14960062135578826456254669380, −2.56371674810102984064330254716, −0.870613339406615733549860978518, 0,
0.870613339406615733549860978518, 2.56371674810102984064330254716, 3.14960062135578826456254669380, 4.63022075865145605392796774799, 5.26560035815040153949942892364, 6.21597557358647333645832838897, 6.90647216259293069350844450107, 7.77979793022761275258522686715, 8.812633765976544954885098670577
