L(s) = 1 | − 2·4-s − 3·5-s − 7-s + 4·13-s + 4·16-s − 6·17-s − 2·19-s + 6·20-s − 3·23-s + 4·25-s + 2·28-s − 6·29-s + 5·31-s + 3·35-s + 11·37-s + 6·41-s − 8·43-s + 49-s − 8·52-s + 6·53-s + 9·59-s + 10·61-s − 8·64-s − 12·65-s + 5·67-s + 12·68-s − 9·71-s + ⋯ |
L(s) = 1 | − 4-s − 1.34·5-s − 0.377·7-s + 1.10·13-s + 16-s − 1.45·17-s − 0.458·19-s + 1.34·20-s − 0.625·23-s + 4/5·25-s + 0.377·28-s − 1.11·29-s + 0.898·31-s + 0.507·35-s + 1.80·37-s + 0.937·41-s − 1.21·43-s + 1/7·49-s − 1.10·52-s + 0.824·53-s + 1.17·59-s + 1.28·61-s − 64-s − 1.48·65-s + 0.610·67-s + 1.45·68-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.79780189933062788077562645237, −6.81062854234304516450450908396, −6.19954355560081421436202244959, −5.35059430166220814299615131856, −4.31297732989480839195031476377, −4.09981281561803996499645678286, −3.45131480487179814577542455626, −2.33532150394146050566081783680, −0.896810126936371404627589617928, 0,
0.896810126936371404627589617928, 2.33532150394146050566081783680, 3.45131480487179814577542455626, 4.09981281561803996499645678286, 4.31297732989480839195031476377, 5.35059430166220814299615131856, 6.19954355560081421436202244959, 6.81062854234304516450450908396, 7.79780189933062788077562645237