L(s) = 1 | + 2·5-s + 2·7-s − 2·11-s − 6·17-s + 6·19-s − 8·23-s − 25-s − 2·29-s − 10·31-s + 4·35-s + 6·37-s − 6·41-s + 4·43-s − 2·47-s − 3·49-s − 6·53-s − 4·55-s − 10·59-s − 2·61-s − 10·67-s + 10·71-s − 2·73-s − 4·77-s − 4·79-s − 6·83-s − 12·85-s − 6·89-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.603·11-s − 1.45·17-s + 1.37·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s − 1.79·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.291·47-s − 3/7·49-s − 0.824·53-s − 0.539·55-s − 1.30·59-s − 0.256·61-s − 1.22·67-s + 1.18·71-s − 0.234·73-s − 0.455·77-s − 0.450·79-s − 0.658·83-s − 1.30·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.75278361549181880377727071254, −7.08998532903080357642556758525, −6.10014918731261564995074453282, −5.64953725554715039802248200782, −4.89233148520829001479133968847, −4.16337264345952282915929548204, −3.12141642932320178028594426715, −2.09132246047893191349158801492, −1.63392338428425511982320095775, 0,
1.63392338428425511982320095775, 2.09132246047893191349158801492, 3.12141642932320178028594426715, 4.16337264345952282915929548204, 4.89233148520829001479133968847, 5.64953725554715039802248200782, 6.10014918731261564995074453282, 7.08998532903080357642556758525, 7.75278361549181880377727071254