L(s) = 1 | + 4·7-s + 4·11-s − 2·13-s − 2·17-s − 19-s + 8·23-s + 6·29-s − 4·31-s − 10·37-s + 2·41-s − 12·43-s + 9·49-s − 6·53-s + 10·61-s + 4·67-s − 8·71-s − 2·73-s + 16·77-s + 12·79-s − 8·83-s − 6·89-s − 8·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.229·19-s + 1.66·23-s + 1.11·29-s − 0.718·31-s − 1.64·37-s + 0.312·41-s − 1.82·43-s + 9/7·49-s − 0.824·53-s + 1.28·61-s + 0.488·67-s − 0.949·71-s − 0.234·73-s + 1.82·77-s + 1.35·79-s − 0.878·83-s − 0.635·89-s − 0.838·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.07996561827630, −12.30047473925009, −12.13025590699145, −11.57456238520823, −11.18259797220070, −10.82723785863870, −10.36000923175387, −9.677306453418730, −9.284196571342833, −8.704020920307855, −8.412463266967034, −8.033398897338699, −7.211899922233409, −6.897086587667446, −6.644934148020107, −5.800239782180480, −5.164276530311820, −4.943110127129954, −4.422879433694834, −3.886927019470221, −3.253797827237839, −2.647275341221118, −1.894978020994035, −1.508213598450835, −0.9680686464382859, 0,
0.9680686464382859, 1.508213598450835, 1.894978020994035, 2.647275341221118, 3.253797827237839, 3.886927019470221, 4.422879433694834, 4.943110127129954, 5.164276530311820, 5.800239782180480, 6.644934148020107, 6.897086587667446, 7.211899922233409, 8.033398897338699, 8.412463266967034, 8.704020920307855, 9.284196571342833, 9.677306453418730, 10.36000923175387, 10.82723785863870, 11.18259797220070, 11.57456238520823, 12.13025590699145, 12.30047473925009, 13.07996561827630