L(s) = 1 | + (−0.5 + 0.866i)4-s + 7-s − 13-s + (−0.499 − 0.866i)16-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + 49-s + (0.5 − 0.866i)52-s + (0.5 + 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)4-s + 7-s − 13-s + (−0.499 − 0.866i)16-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + 49-s + (0.5 − 0.866i)52-s + (0.5 + 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6520774674\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6520774674\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−12.95712025298803857390028926301, −11.84183935124613510867712083098, −11.18394374517780049218032169032, −9.771200136271633034970017040564, −8.752839438830055882330236675064, −7.88639050962153812660136432658, −6.92610513305897376346028673064, −5.11203554069154677044449361215, −4.26297223858260436199025984889, −2.55939948811483532521196998680,
1.91255076604218535700168097614, 4.20744014396295569953301800008, 5.19214167103579096857904269941, 6.30087174016502298916855520295, 7.80499230711910464163221183475, 8.703775877156780684159818138802, 9.962045875563006616284826861808, 10.56154615413713456608029130908, 11.76185879594565457888328419177, 12.69411025483152826804369717737