| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·13-s − 14-s + 16-s − 2·17-s + 6·23-s − 4·26-s − 28-s + 2·31-s + 32-s − 2·34-s + 8·37-s + 2·41-s − 4·43-s + 6·46-s − 8·47-s + 49-s − 4·52-s + 6·53-s − 56-s − 2·61-s + 2·62-s + 64-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 1.25·23-s − 0.784·26-s − 0.188·28-s + 0.359·31-s + 0.176·32-s − 0.342·34-s + 1.31·37-s + 0.312·41-s − 0.609·43-s + 0.884·46-s − 1.16·47-s + 1/7·49-s − 0.554·52-s + 0.824·53-s − 0.133·56-s − 0.256·61-s + 0.254·62-s + 1/8·64-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.61970366950210, −12.52366522326611, −11.71262569919632, −11.45353392939605, −11.08947878441555, −10.40283586712420, −10.09201383830434, −9.500766337949573, −9.252206053156211, −8.535130644525781, −8.112806188527904, −7.528415489529910, −7.090594400046786, −6.669359737625782, −6.306685539437023, −5.596177288729095, −5.238235199895622, −4.680715992167299, −4.335160579611774, −3.725127704523837, −3.077511269019327, −2.691532100159282, −2.242211643737162, −1.482014468888316, −0.7901485347547286, 0,
0.7901485347547286, 1.482014468888316, 2.242211643737162, 2.691532100159282, 3.077511269019327, 3.725127704523837, 4.335160579611774, 4.680715992167299, 5.238235199895622, 5.596177288729095, 6.306685539437023, 6.669359737625782, 7.090594400046786, 7.528415489529910, 8.112806188527904, 8.535130644525781, 9.252206053156211, 9.500766337949573, 10.09201383830434, 10.40283586712420, 11.08947878441555, 11.45353392939605, 11.71262569919632, 12.52366522326611, 12.61970366950210
