L(s) = 1 | + 2-s + 4-s + 8-s − 13-s + 16-s + 2·17-s − 8·19-s − 4·23-s − 26-s + 2·29-s − 4·31-s + 32-s + 2·34-s − 6·37-s − 8·38-s − 10·41-s − 4·46-s + 8·47-s − 7·49-s − 52-s + 6·53-s + 2·58-s − 8·59-s − 2·61-s − 4·62-s + 64-s − 4·67-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 1.83·19-s − 0.834·23-s − 0.196·26-s + 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.986·37-s − 1.29·38-s − 1.56·41-s − 0.589·46-s + 1.16·47-s − 49-s − 0.138·52-s + 0.824·53-s + 0.262·58-s − 1.04·59-s − 0.256·61-s − 0.508·62-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.67166429832890002882383765523, −6.88129506287611830331283034917, −6.29137854801807922792370058555, −5.56728720663395897693496802886, −4.80814816812391826250906168460, −4.08273503358194901989986669834, −3.37599360054645049754720862286, −2.37806233738479954556599834298, −1.62154349299587194944953326977, 0,
1.62154349299587194944953326977, 2.37806233738479954556599834298, 3.37599360054645049754720862286, 4.08273503358194901989986669834, 4.80814816812391826250906168460, 5.56728720663395897693496802886, 6.29137854801807922792370058555, 6.88129506287611830331283034917, 7.67166429832890002882383765523