| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 2·13-s + 15-s − 6·17-s − 4·19-s − 2·21-s + 8·23-s + 25-s + 27-s − 4·31-s − 2·35-s + 10·37-s + 2·39-s + 4·41-s + 2·43-s + 45-s + 8·47-s − 3·49-s − 6·51-s + 6·53-s − 4·57-s − 10·59-s − 6·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.338·35-s + 1.64·37-s + 0.320·39-s + 0.624·41-s + 0.304·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 1.30·59-s − 0.768·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.98229346840241, −14.61013244678885, −13.94895562245432, −13.31015479908361, −13.00151614969238, −12.84762059673378, −11.97199242603204, −11.21113874729035, −10.78036869310119, −10.43181741520199, −9.514106030972744, −9.197637157342190, −8.834275911953420, −8.259151591707149, −7.376562676385633, −7.034042694149715, −6.261529664440651, −6.032318106643785, −5.137779901629992, −4.293071304425473, −4.075161949831065, −2.926638140665006, −2.769463400423026, −1.886236338341759, −1.084154370347009, 0,
1.084154370347009, 1.886236338341759, 2.769463400423026, 2.926638140665006, 4.075161949831065, 4.293071304425473, 5.137779901629992, 6.032318106643785, 6.261529664440651, 7.034042694149715, 7.376562676385633, 8.259151591707149, 8.834275911953420, 9.197637157342190, 9.514106030972744, 10.43181741520199, 10.78036869310119, 11.21113874729035, 11.97199242603204, 12.84762059673378, 13.00151614969238, 13.31015479908361, 13.94895562245432, 14.61013244678885, 14.98229346840241
