L(s) = 1 | − 5-s + 4·7-s − 4·11-s − 2·13-s + 6·17-s + 6·19-s − 6·23-s + 25-s + 2·29-s − 4·31-s − 4·35-s − 8·37-s + 8·41-s + 43-s − 6·47-s + 9·49-s − 6·53-s + 4·55-s + 10·61-s + 2·65-s − 12·67-s + 16·71-s + 16·73-s − 16·77-s − 4·79-s − 6·83-s − 6·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.676·35-s − 1.31·37-s + 1.24·41-s + 0.152·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 1.28·61-s + 0.248·65-s − 1.46·67-s + 1.89·71-s + 1.87·73-s − 1.82·77-s − 0.450·79-s − 0.658·83-s − 0.650·85-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
\(L(1)\) |
\(\approx\) |
\(2.348632723\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.348632723\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.84928192807026, −12.88874592292545, −12.48349764763358, −12.08778976531375, −11.61074693460181, −11.15603894499708, −10.71572159396318, −10.09306587891022, −9.790730415809786, −9.164735619746109, −8.323873544075063, −8.062061529518317, −7.698186229416579, −7.361318414691698, −6.691904134617341, −5.628418050113936, −5.516835678720278, −4.978849063708268, −4.512198908670473, −3.731806612282787, −3.245232946952862, −2.530521336211712, −1.898249220672363, −1.257076362405489, −0.4848786793911440,
0.4848786793911440, 1.257076362405489, 1.898249220672363, 2.530521336211712, 3.245232946952862, 3.731806612282787, 4.512198908670473, 4.978849063708268, 5.516835678720278, 5.628418050113936, 6.691904134617341, 7.361318414691698, 7.698186229416579, 8.062061529518317, 8.323873544075063, 9.164735619746109, 9.790730415809786, 10.09306587891022, 10.71572159396318, 11.15603894499708, 11.61074693460181, 12.08778976531375, 12.48349764763358, 12.88874592292545, 13.84928192807026