L(s) = 1 | − 3-s − 7-s − 2·9-s − 6·11-s + 5·13-s − 3·17-s + 19-s + 21-s + 3·23-s + 5·27-s − 9·29-s + 4·31-s + 6·33-s + 2·37-s − 5·39-s − 8·43-s − 6·49-s + 3·51-s − 3·53-s − 57-s + 9·59-s + 10·61-s + 2·63-s − 5·67-s − 3·69-s + 6·71-s + 7·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.80·11-s + 1.38·13-s − 0.727·17-s + 0.229·19-s + 0.218·21-s + 0.625·23-s + 0.962·27-s − 1.67·29-s + 0.718·31-s + 1.04·33-s + 0.328·37-s − 0.800·39-s − 1.21·43-s − 6/7·49-s + 0.420·51-s − 0.412·53-s − 0.132·57-s + 1.17·59-s + 1.28·61-s + 0.251·63-s − 0.610·67-s − 0.361·69-s + 0.712·71-s + 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−15.43880003325881, −15.03057519218786, −14.31576758988019, −13.53247556629647, −13.23186655532658, −12.95093099069880, −12.19296552678570, −11.45914007216986, −11.02706676489048, −10.84720406152769, −10.05732054034554, −9.544983433385998, −8.746826995159473, −8.330954849814055, −7.826507880517052, −7.032812774283390, −6.417046990580210, −5.941883400857638, −5.207992974420034, −5.027443002960905, −3.950251637679465, −3.300775693458464, −2.685171382196544, −1.922126933858216, −0.7832900630647673, 0,
0.7832900630647673, 1.922126933858216, 2.685171382196544, 3.300775693458464, 3.950251637679465, 5.027443002960905, 5.207992974420034, 5.941883400857638, 6.417046990580210, 7.032812774283390, 7.826507880517052, 8.330954849814055, 8.746826995159473, 9.544983433385998, 10.05732054034554, 10.84720406152769, 11.02706676489048, 11.45914007216986, 12.19296552678570, 12.95093099069880, 13.23186655532658, 13.53247556629647, 14.31576758988019, 15.03057519218786, 15.43880003325881