L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 13-s − 2·14-s + 16-s + 2·19-s + 20-s + 6·23-s + 25-s − 26-s + 2·28-s − 4·31-s − 32-s + 2·35-s + 2·37-s − 2·38-s − 40-s + 6·41-s − 4·43-s − 6·46-s − 3·49-s − 50-s + 52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.458·19-s + 0.223·20-s + 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.377·28-s − 0.718·31-s − 0.176·32-s + 0.338·35-s + 0.328·37-s − 0.324·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s − 0.884·46-s − 3/7·49-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.455540702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.455540702\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.610307633874040285371220817619, −9.050231564737947469990813268837, −8.195553408380404831920993641146, −7.44200053996330116321929479527, −6.58298325388265669739783786821, −5.59531005218261041497813692032, −4.75705494829356928996463504748, −3.39519945126596367924943215518, −2.19648591121885948588699453544, −1.07994497826633863377468377161,
1.07994497826633863377468377161, 2.19648591121885948588699453544, 3.39519945126596367924943215518, 4.75705494829356928996463504748, 5.59531005218261041497813692032, 6.58298325388265669739783786821, 7.44200053996330116321929479527, 8.195553408380404831920993641146, 9.050231564737947469990813268837, 9.610307633874040285371220817619