L(s) = 1 | − 5-s + 7-s − 11-s + 3·13-s − 7·19-s + 6·23-s − 4·25-s + 9·29-s − 35-s − 3·37-s − 8·41-s − 10·43-s + 3·47-s + 49-s − 6·53-s + 55-s + 7·59-s + 10·61-s − 3·65-s + 3·67-s − 8·71-s − 7·73-s − 77-s − 8·79-s + 6·89-s + 3·91-s + 7·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.832·13-s − 1.60·19-s + 1.25·23-s − 4/5·25-s + 1.67·29-s − 0.169·35-s − 0.493·37-s − 1.24·41-s − 1.52·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s + 0.911·59-s + 1.28·61-s − 0.372·65-s + 0.366·67-s − 0.949·71-s − 0.819·73-s − 0.113·77-s − 0.900·79-s + 0.635·89-s + 0.314·91-s + 0.718·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.77140263316671, −16.12456623209989, −15.60680914610460, −15.09802657201201, −14.64071720197185, −13.86073395335790, −13.32370902642794, −12.84283673924929, −12.11548372221718, −11.54314804839810, −11.04265852398475, −10.38119236616632, −9.969226434896554, −8.885078023630616, −8.440873711056017, −8.143191158944056, −7.115129032808001, −6.665793645196973, −5.943711568351380, −5.084861630673387, −4.522471790259658, −3.770146464344528, −3.038739524443087, −2.103822624736416, −1.199128633640421, 0,
1.199128633640421, 2.103822624736416, 3.038739524443087, 3.770146464344528, 4.522471790259658, 5.084861630673387, 5.943711568351380, 6.665793645196973, 7.115129032808001, 8.143191158944056, 8.440873711056017, 8.885078023630616, 9.969226434896554, 10.38119236616632, 11.04265852398475, 11.54314804839810, 12.11548372221718, 12.84283673924929, 13.32370902642794, 13.86073395335790, 14.64071720197185, 15.09802657201201, 15.60680914610460, 16.12456623209989, 16.77140263316671