| L(s) = 1 | − 2·7-s − 4·11-s − 6·13-s − 5·17-s − 6·19-s + 3·23-s + 8·29-s + 5·31-s − 6·37-s + 3·41-s − 43-s − 47-s − 3·49-s + 9·53-s − 7·59-s + 67-s − 12·71-s − 2·73-s + 8·77-s − 79-s + 12·83-s − 6·89-s + 12·91-s − 9·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s − 1.66·13-s − 1.21·17-s − 1.37·19-s + 0.625·23-s + 1.48·29-s + 0.898·31-s − 0.986·37-s + 0.468·41-s − 0.152·43-s − 0.145·47-s − 3/7·49-s + 1.23·53-s − 0.911·59-s + 0.122·67-s − 1.42·71-s − 0.234·73-s + 0.911·77-s − 0.112·79-s + 1.31·83-s − 0.635·89-s + 1.25·91-s − 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.45221682169416, −13.10549408880504, −12.57108180015584, −12.21631426104416, −11.83203925701478, −10.93052969960067, −10.71832459792518, −10.15338083015033, −9.872067189027927, −9.226478421941094, −8.705765593899726, −8.279210696777850, −7.735009492779298, −7.078306800963910, −6.738398840005085, −6.327452945628458, −5.588728382222835, −5.024210762018200, −4.550261270416991, −4.216922056472692, −3.216304256313444, −2.637691202028462, −2.490978643338765, −1.680852398542932, −0.5311185115310144, 0,
0.5311185115310144, 1.680852398542932, 2.490978643338765, 2.637691202028462, 3.216304256313444, 4.216922056472692, 4.550261270416991, 5.024210762018200, 5.588728382222835, 6.327452945628458, 6.738398840005085, 7.078306800963910, 7.735009492779298, 8.279210696777850, 8.705765593899726, 9.226478421941094, 9.872067189027927, 10.15338083015033, 10.71832459792518, 10.93052969960067, 11.83203925701478, 12.21631426104416, 12.57108180015584, 13.10549408880504, 13.45221682169416
