| L(s) = 1 | + 2·11-s − 3·13-s + 5·17-s + 4·19-s + 5·23-s − 9·29-s − 5·31-s − 10·37-s + 41-s + 3·43-s + 12·47-s + 11·53-s − 5·59-s − 9·61-s − 4·67-s − 4·71-s − 6·73-s − 14·79-s + 5·83-s − 6·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 0.832·13-s + 1.21·17-s + 0.917·19-s + 1.04·23-s − 1.67·29-s − 0.898·31-s − 1.64·37-s + 0.156·41-s + 0.457·43-s + 1.75·47-s + 1.51·53-s − 0.650·59-s − 1.15·61-s − 0.488·67-s − 0.474·71-s − 0.702·73-s − 1.57·79-s + 0.548·83-s − 0.635·89-s − 1.82·97-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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.62000497749165, −12.81180217070173, −12.41164606335934, −12.07841767336880, −11.59973370673437, −11.08540057949842, −10.51973217190558, −10.16962816419925, −9.498598143544700, −9.130152336541989, −8.878731277544457, −8.047362701306390, −7.475870920867386, −7.169999822269781, −6.914973827811631, −5.818102327430798, −5.608411971414781, −5.246949349427502, −4.407041549843782, −3.992453375681891, −3.216471664128116, −3.019452739798014, −2.076370883903023, −1.516804972516911, −0.8748995300066667, 0,
0.8748995300066667, 1.516804972516911, 2.076370883903023, 3.019452739798014, 3.216471664128116, 3.992453375681891, 4.407041549843782, 5.246949349427502, 5.608411971414781, 5.818102327430798, 6.914973827811631, 7.169999822269781, 7.475870920867386, 8.047362701306390, 8.878731277544457, 9.130152336541989, 9.498598143544700, 10.16962816419925, 10.51973217190558, 11.08540057949842, 11.59973370673437, 12.07841767336880, 12.41164606335934, 12.81180217070173, 13.62000497749165
