L(s) = 1 | + 2·5-s + 2·7-s − 2·13-s + 4·17-s − 6·19-s − 25-s + 8·29-s − 8·31-s + 4·35-s − 10·37-s + 8·41-s − 2·43-s + 8·47-s − 3·49-s − 2·53-s + 12·59-s + 10·61-s − 4·65-s − 12·67-s − 8·71-s − 6·73-s + 2·79-s − 16·83-s + 8·85-s + 14·89-s − 4·91-s − 12·95-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.554·13-s + 0.970·17-s − 1.37·19-s − 1/5·25-s + 1.48·29-s − 1.43·31-s + 0.676·35-s − 1.64·37-s + 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 1.56·59-s + 1.28·61-s − 0.496·65-s − 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.225·79-s − 1.75·83-s + 0.867·85-s + 1.48·89-s − 0.419·91-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.38227283306540, −14.14312381638523, −13.26418446362410, −13.04877112602439, −12.28960642639212, −12.04799364403350, −11.35807937145454, −10.78019826774671, −10.22738489803651, −10.06366797436295, −9.302575667640989, −8.670721471723700, −8.473587248237279, −7.561887722649132, −7.316580586810413, −6.520624637549553, −6.020331846436753, −5.424445709979690, −5.046023736961219, −4.329122974347763, −3.780103610380525, −2.918141752344186, −2.262332492860653, −1.779831797588810, −1.075130590742313, 0,
1.075130590742313, 1.779831797588810, 2.262332492860653, 2.918141752344186, 3.780103610380525, 4.329122974347763, 5.046023736961219, 5.424445709979690, 6.020331846436753, 6.520624637549553, 7.316580586810413, 7.561887722649132, 8.473587248237279, 8.670721471723700, 9.302575667640989, 10.06366797436295, 10.22738489803651, 10.78019826774671, 11.35807937145454, 12.04799364403350, 12.28960642639212, 13.04877112602439, 13.26418446362410, 14.14312381638523, 14.38227283306540