| L(s) = 1 | − 2-s + 4-s + 5-s − 5·7-s − 8-s − 10-s − 11-s − 4·13-s + 5·14-s + 16-s − 6·17-s + 3·19-s + 20-s + 22-s − 7·23-s + 25-s + 4·26-s − 5·28-s + 8·29-s − 31-s − 32-s + 6·34-s − 5·35-s + 8·37-s − 3·38-s − 40-s + 43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 1.10·13-s + 1.33·14-s + 1/4·16-s − 1.45·17-s + 0.688·19-s + 0.223·20-s + 0.213·22-s − 1.45·23-s + 1/5·25-s + 0.784·26-s − 0.944·28-s + 1.48·29-s − 0.179·31-s − 0.176·32-s + 1.02·34-s − 0.845·35-s + 1.31·37-s − 0.486·38-s − 0.158·40-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6572632530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6572632530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.041248707027259705161441758606, −8.089214143925380338016885102443, −7.24746775187893654259537718296, −6.49241448966187991756309763333, −6.12654788155122086648290255972, −5.00368084780327692093630652847, −3.89028041409452054406331400388, −2.78745042395360105363916886348, −2.27220021818424651939667936571, −0.52006726175440794425980175575,
0.52006726175440794425980175575, 2.27220021818424651939667936571, 2.78745042395360105363916886348, 3.89028041409452054406331400388, 5.00368084780327692093630652847, 6.12654788155122086648290255972, 6.49241448966187991756309763333, 7.24746775187893654259537718296, 8.089214143925380338016885102443, 9.041248707027259705161441758606
