L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s − 4·11-s − 2·13-s + 2·14-s − 4·16-s + 3·17-s − 8·19-s − 2·20-s + 8·22-s − 6·23-s − 4·25-s + 4·26-s − 2·28-s − 4·29-s + 6·31-s + 8·32-s − 6·34-s + 35-s − 3·37-s + 16·38-s + 41-s + 11·43-s − 8·44-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s − 1.20·11-s − 0.554·13-s + 0.534·14-s − 16-s + 0.727·17-s − 1.83·19-s − 0.447·20-s + 1.70·22-s − 1.25·23-s − 4/5·25-s + 0.784·26-s − 0.377·28-s − 0.742·29-s + 1.07·31-s + 1.41·32-s − 1.02·34-s + 0.169·35-s − 0.493·37-s + 2.59·38-s + 0.156·41-s + 1.67·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−11.89714523984240440162857130644, −10.60299964121421993605621307276, −10.17243898255102214941018911651, −9.039863529492895542714101940401, −8.011472352484350129239935128289, −7.43939934466159987675241180651, −5.97786895192084747528753482689, −4.28481394661801806092561213309, −2.34130595001443635884001685756, 0,
2.34130595001443635884001685756, 4.28481394661801806092561213309, 5.97786895192084747528753482689, 7.43939934466159987675241180651, 8.011472352484350129239935128289, 9.039863529492895542714101940401, 10.17243898255102214941018911651, 10.60299964121421993605621307276, 11.89714523984240440162857130644