L(s) = 1 | − 3-s − 3·7-s − 2·9-s + 3·17-s + 3·19-s + 3·21-s + 5·27-s − 9·29-s + 5·31-s + 11·37-s − 6·41-s + 12·43-s + 6·47-s + 2·49-s − 3·51-s + 3·53-s − 3·57-s − 12·59-s − 15·61-s + 6·63-s + 4·67-s + 3·71-s − 6·73-s + 12·79-s + 81-s − 12·83-s + 9·87-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s + 0.727·17-s + 0.688·19-s + 0.654·21-s + 0.962·27-s − 1.67·29-s + 0.898·31-s + 1.80·37-s − 0.937·41-s + 1.82·43-s + 0.875·47-s + 2/7·49-s − 0.420·51-s + 0.412·53-s − 0.397·57-s − 1.56·59-s − 1.92·61-s + 0.755·63-s + 0.488·67-s + 0.356·71-s − 0.702·73-s + 1.35·79-s + 1/9·81-s − 1.31·83-s + 0.964·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.136005489\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.136005489\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−14.53239699572642, −14.03362560918499, −13.45386155019806, −13.04913564479953, −12.35568430417901, −12.07570431966516, −11.52386427853245, −10.88064673413739, −10.56386285044391, −9.757472747153833, −9.418605426925909, −8.999221066232748, −8.160736809209974, −7.585710012912059, −7.179160702573264, −6.230575400694287, −6.065485169100975, −5.563343597045158, −4.839024087254265, −4.155726575406667, −3.355289372479657, −2.984921833746880, −2.244470485011614, −1.136971493523081, −0.4386737609681390,
0.4386737609681390, 1.136971493523081, 2.244470485011614, 2.984921833746880, 3.355289372479657, 4.155726575406667, 4.839024087254265, 5.563343597045158, 6.065485169100975, 6.230575400694287, 7.179160702573264, 7.585710012912059, 8.160736809209974, 8.999221066232748, 9.418605426925909, 9.757472747153833, 10.56386285044391, 10.88064673413739, 11.52386427853245, 12.07570431966516, 12.35568430417901, 13.04913564479953, 13.45386155019806, 14.03362560918499, 14.53239699572642