L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s − 4·7-s − 3·8-s + 9-s + 2·10-s + 4·11-s + 12-s + 13-s − 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s − 2·20-s + 4·21-s + 4·22-s + 3·24-s − 25-s + 26-s − 27-s + 4·28-s − 10·29-s − 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s + 0.872·21-s + 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.85·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8266878506\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8266878506\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.41358335985935193688740154223, −14.93804365278039683919899557472, −13.67450328613745713035176093198, −12.95684499975161998376414511217, −11.85387650647454943465550811155, −9.952802346640542280811898231944, −9.192740554674130435254875131876, −6.54556545042783689480045128663, −5.65486780279705070512904066158, −3.71205447024015551816570926916,
3.71205447024015551816570926916, 5.65486780279705070512904066158, 6.54556545042783689480045128663, 9.192740554674130435254875131876, 9.952802346640542280811898231944, 11.85387650647454943465550811155, 12.95684499975161998376414511217, 13.67450328613745713035176093198, 14.93804365278039683919899557472, 16.41358335985935193688740154223