L(s) = 1 | + 3-s + 3·7-s − 2·9-s − 3·13-s + 3·17-s + 3·21-s − 9·23-s − 5·25-s − 5·27-s + 9·29-s + 6·31-s − 6·37-s − 3·39-s + 6·41-s + 8·43-s + 2·49-s + 3·51-s + 9·53-s − 3·59-s + 6·61-s − 6·63-s + 5·67-s − 9·69-s − 11·73-s − 5·75-s − 12·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s − 0.832·13-s + 0.727·17-s + 0.654·21-s − 1.87·23-s − 25-s − 0.962·27-s + 1.67·29-s + 1.07·31-s − 0.986·37-s − 0.480·39-s + 0.937·41-s + 1.21·43-s + 2/7·49-s + 0.420·51-s + 1.23·53-s − 0.390·59-s + 0.768·61-s − 0.755·63-s + 0.610·67-s − 1.08·69-s − 1.28·73-s − 0.577·75-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.11381646491932, −13.96353985679745, −13.22694333889856, −12.45547875098089, −12.02280881611697, −11.71801422053097, −11.31249591602776, −10.41547532122158, −10.16945002201389, −9.693825935141018, −8.953003868221744, −8.510635117388502, −7.956573550671330, −7.825220297866853, −7.178088211904552, −6.397636280923584, −5.751853737294801, −5.461821556385300, −4.651631235804815, −4.228230574766160, −3.635805514645603, −2.748963856109943, −2.424880025046159, −1.767375214876645, −0.9691906834699114, 0,
0.9691906834699114, 1.767375214876645, 2.424880025046159, 2.748963856109943, 3.635805514645603, 4.228230574766160, 4.651631235804815, 5.461821556385300, 5.751853737294801, 6.397636280923584, 7.178088211904552, 7.825220297866853, 7.956573550671330, 8.510635117388502, 8.953003868221744, 9.693825935141018, 10.16945002201389, 10.41547532122158, 11.31249591602776, 11.71801422053097, 12.02280881611697, 12.45547875098089, 13.22694333889856, 13.96353985679745, 14.11381646491932