L(s) = 1 | − 4·11-s − 2·13-s − 2·17-s − 4·19-s + 10·29-s − 6·37-s − 6·41-s − 4·43-s − 8·47-s + 6·53-s + 4·59-s + 10·61-s + 4·67-s − 16·71-s − 14·73-s − 8·79-s − 4·83-s + 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.85·29-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 0.824·53-s + 0.520·59-s + 1.28·61-s + 0.488·67-s − 1.89·71-s − 1.63·73-s − 0.900·79-s − 0.439·83-s + 1.05·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.22566949556918, −13.16769914391548, −12.43870255164297, −12.04842238431525, −11.57087694557794, −11.05265454074421, −10.38730594424121, −10.16299811504157, −9.901034105206646, −8.956807800482822, −8.532464115914275, −8.317521920239975, −7.662428162899163, −7.013481369348967, −6.766288115197891, −6.127128966283690, −5.490081836575637, −5.031652837482059, −4.555159087835085, −4.087922903016510, −3.138810872392983, −2.896644869276118, −2.138556497366070, −1.696104535061678, −0.6498230182312536, 0,
0.6498230182312536, 1.696104535061678, 2.138556497366070, 2.896644869276118, 3.138810872392983, 4.087922903016510, 4.555159087835085, 5.031652837482059, 5.490081836575637, 6.127128966283690, 6.766288115197891, 7.013481369348967, 7.662428162899163, 8.317521920239975, 8.532464115914275, 8.956807800482822, 9.901034105206646, 10.16299811504157, 10.38730594424121, 11.05265454074421, 11.57087694557794, 12.04842238431525, 12.43870255164297, 13.16769914391548, 13.22566949556918