| L(s) = 1 | + 5-s − 2·7-s + 11-s + 7·13-s − 3·17-s − 6·19-s + 5·23-s + 25-s + 6·29-s − 7·31-s − 2·35-s + 10·37-s − 5·41-s − 43-s + 4·47-s − 3·49-s + 53-s + 55-s − 8·59-s + 8·61-s + 7·65-s + 3·67-s − 8·71-s − 8·73-s − 2·77-s + 8·79-s − 9·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.301·11-s + 1.94·13-s − 0.727·17-s − 1.37·19-s + 1.04·23-s + 1/5·25-s + 1.11·29-s − 1.25·31-s − 0.338·35-s + 1.64·37-s − 0.780·41-s − 0.152·43-s + 0.583·47-s − 3/7·49-s + 0.137·53-s + 0.134·55-s − 1.04·59-s + 1.02·61-s + 0.868·65-s + 0.366·67-s − 0.949·71-s − 0.936·73-s − 0.227·77-s + 0.900·79-s − 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−13.54717205594086, −13.29711286940597, −12.88699584093996, −12.58971261777198, −11.77273834315117, −11.19791979892067, −10.92596552992182, −10.46604702934493, −9.890729279865820, −9.302875816861975, −8.820519641750182, −8.592671236481638, −8.012802331332471, −7.135723391683323, −6.719337606525343, −6.217757037384577, −6.016002117337035, −5.288008031530511, −4.498748057009852, −4.110243709793354, −3.462714129138098, −2.935510519023517, −2.262594335456931, −1.522069615304752, −0.9423544067888743, 0,
0.9423544067888743, 1.522069615304752, 2.262594335456931, 2.935510519023517, 3.462714129138098, 4.110243709793354, 4.498748057009852, 5.288008031530511, 6.016002117337035, 6.217757037384577, 6.719337606525343, 7.135723391683323, 8.012802331332471, 8.592671236481638, 8.820519641750182, 9.302875816861975, 9.890729279865820, 10.46604702934493, 10.92596552992182, 11.19791979892067, 11.77273834315117, 12.58971261777198, 12.88699584093996, 13.29711286940597, 13.54717205594086
