L(s) = 1 | + 2-s + 4-s + 4·5-s − 4·7-s + 8-s − 3·9-s + 4·10-s + 4·11-s − 4·13-s − 4·14-s + 16-s + 6·17-s − 3·18-s − 6·19-s + 4·20-s + 4·22-s − 4·23-s + 11·25-s − 4·26-s − 4·28-s + 32-s + 6·34-s − 16·35-s − 3·36-s − 8·37-s − 6·38-s + 4·40-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 1.51·7-s + 0.353·8-s − 9-s + 1.26·10-s + 1.20·11-s − 1.10·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.37·19-s + 0.894·20-s + 0.852·22-s − 0.834·23-s + 11/5·25-s − 0.784·26-s − 0.755·28-s + 0.176·32-s + 1.02·34-s − 2.70·35-s − 1/2·36-s − 1.31·37-s − 0.973·38-s + 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.887548874\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.887548874\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
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
−12.62101468550581982510626400546, −11.91392388902353971094597148292, −10.24883442029425890282292876544, −9.808731036004418938550881898907, −8.767999538368381255541931230345, −6.79502419315262861884017444820, −6.16068938165576146811569566993, −5.33519313821748327224317218590, −3.46279750097009321792818482789, −2.22991181864696470461280833479,
2.22991181864696470461280833479, 3.46279750097009321792818482789, 5.33519313821748327224317218590, 6.16068938165576146811569566993, 6.79502419315262861884017444820, 8.767999538368381255541931230345, 9.808731036004418938550881898907, 10.24883442029425890282292876544, 11.91392388902353971094597148292, 12.62101468550581982510626400546