L(s) = 1 | − 3·3-s + 5-s + 6·9-s − 3·13-s − 3·15-s − 17-s + 6·19-s + 6·23-s + 25-s − 9·27-s + 9·29-s + 4·31-s + 2·37-s + 9·39-s − 4·41-s − 10·43-s + 6·45-s + 47-s + 3·51-s + 4·53-s − 18·57-s + 8·59-s − 8·61-s − 3·65-s + 12·67-s − 18·69-s + 8·71-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s − 0.832·13-s − 0.774·15-s − 0.242·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s + 0.718·31-s + 0.328·37-s + 1.44·39-s − 0.624·41-s − 1.52·43-s + 0.894·45-s + 0.145·47-s + 0.420·51-s + 0.549·53-s − 2.38·57-s + 1.04·59-s − 1.02·61-s − 0.372·65-s + 1.46·67-s − 2.16·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.71082697874367, −13.27602074623255, −12.63073121593453, −12.37486287881056, −11.73883520536276, −11.49349279542079, −11.07590367631564, −10.24857796981473, −10.12944115436754, −9.724105880309859, −9.013939936760954, −8.422713677922032, −7.736002954187432, −7.065177411932629, −6.781490601146915, −6.371025204799398, −5.642674205507643, −5.253812290451795, −4.812332434515617, −4.498634265750719, −3.515730883095756, −2.880761211723664, −2.188206954989884, −1.190809856243946, −0.9236783324041348, 0,
0.9236783324041348, 1.190809856243946, 2.188206954989884, 2.880761211723664, 3.515730883095756, 4.498634265750719, 4.812332434515617, 5.253812290451795, 5.642674205507643, 6.371025204799398, 6.781490601146915, 7.065177411932629, 7.736002954187432, 8.422713677922032, 9.013939936760954, 9.724105880309859, 10.12944115436754, 10.24857796981473, 11.07590367631564, 11.49349279542079, 11.73883520536276, 12.37486287881056, 12.63073121593453, 13.27602074623255, 13.71082697874367