L(s) = 1 | − 3-s + 7-s − 2·9-s + 13-s + 17-s − 4·19-s − 21-s + 5·27-s − 6·29-s − 31-s − 2·37-s − 39-s − 2·43-s + 6·47-s − 6·49-s − 51-s − 3·53-s + 4·57-s − 6·59-s − 10·61-s − 2·63-s + 4·67-s − 3·71-s − 2·73-s − 79-s + 81-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.277·13-s + 0.242·17-s − 0.917·19-s − 0.218·21-s + 0.962·27-s − 1.11·29-s − 0.179·31-s − 0.328·37-s − 0.160·39-s − 0.304·43-s + 0.875·47-s − 6/7·49-s − 0.140·51-s − 0.412·53-s + 0.529·57-s − 0.781·59-s − 1.28·61-s − 0.251·63-s + 0.488·67-s − 0.356·71-s − 0.234·73-s − 0.112·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.865263655247040843709193930123, −8.222135423899438308933407572488, −7.34275530339690718799566248411, −6.37587539769610241146730858364, −5.72262885421785953868421921516, −4.92891255426859010650275400295, −3.95558898967368997234840864068, −2.84130805973745766678259195577, −1.59165464203817659633355943598, 0,
1.59165464203817659633355943598, 2.84130805973745766678259195577, 3.95558898967368997234840864068, 4.92891255426859010650275400295, 5.72262885421785953868421921516, 6.37587539769610241146730858364, 7.34275530339690718799566248411, 8.222135423899438308933407572488, 8.865263655247040843709193930123