| L(s) = 1 | − 3-s − 2·4-s − 5-s − 2·7-s − 2·9-s − 11-s + 2·12-s − 13-s + 15-s + 4·16-s − 4·17-s + 2·19-s + 2·20-s + 2·21-s + 7·23-s − 4·25-s + 5·27-s + 4·28-s − 2·29-s − 3·31-s + 33-s + 2·35-s + 4·36-s − 11·37-s + 39-s + 10·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s − 0.970·17-s + 0.458·19-s + 0.447·20-s + 0.436·21-s + 1.45·23-s − 4/5·25-s + 0.962·27-s + 0.755·28-s − 0.371·29-s − 0.538·31-s + 0.174·33-s + 0.338·35-s + 2/3·36-s − 1.80·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−12.69739775523992868635283060361, −11.63719964346043465889649554011, −10.62154482736631832225600054915, −9.421092482696968254279932519907, −8.585957272776804427928347593039, −7.21661224338768342466420756206, −5.82135849339387357739621546932, −4.75768041533452306802121891710, −3.29304594675951575785139026233, 0,
3.29304594675951575785139026233, 4.75768041533452306802121891710, 5.82135849339387357739621546932, 7.21661224338768342466420756206, 8.585957272776804427928347593039, 9.421092482696968254279932519907, 10.62154482736631832225600054915, 11.63719964346043465889649554011, 12.69739775523992868635283060361
