L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 4·11-s + 12-s + 2·13-s + 14-s + 16-s + 6·17-s + 18-s + 21-s − 4·22-s + 8·23-s + 24-s + 2·26-s + 27-s + 28-s + 10·29-s − 8·31-s + 32-s − 4·33-s + 6·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.206165521\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.206165521\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11558774468085427099984179567, −8.970038638688739077275471188963, −8.117271450797248149047450674807, −7.49183328013247917757345935069, −6.51937316169267569375873124058, −5.35085092662211501216455986720, −4.81962159294781503660641716359, −3.46413801666508801779103816982, −2.83397636642881070566824261870, −1.42737253017477970910017504740,
1.42737253017477970910017504740, 2.83397636642881070566824261870, 3.46413801666508801779103816982, 4.81962159294781503660641716359, 5.35085092662211501216455986720, 6.51937316169267569375873124058, 7.49183328013247917757345935069, 8.117271450797248149047450674807, 8.970038638688739077275471188963, 10.11558774468085427099984179567