L(s) = 1 | − i·3-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.587 + 0.809i)7-s − 9-s + (0.309 − 0.951i)11-s + (−0.587 + 0.809i)12-s + (−1.53 − 0.5i)13-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.587 − 1.80i)17-s + (−0.951 − 0.309i)20-s + (0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s + i·27-s + (0.951 − 0.309i)28-s + ⋯ |
L(s) = 1 | − i·3-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.587 + 0.809i)7-s − 9-s + (0.309 − 0.951i)11-s + (−0.587 + 0.809i)12-s + (−1.53 − 0.5i)13-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.587 − 1.80i)17-s + (−0.951 − 0.309i)20-s + (0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s + i·27-s + (0.951 − 0.309i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7349561352\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7349561352\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.951 + 0.309i)T \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−9.422738416319886160765323812639, −9.033507192681881536178822232805, −8.205318601298600748956465796085, −6.97769932783474972738217373098, −6.25048476775163456392202889542, −5.38645629302314553933728937523, −4.96298824304510782910092388662, −3.03675516905852479475590265397, −2.20863429100374431803230802104, −0.64427114098009765078116628240,
2.23069044410926968136325685752, 3.43192396783668112536100469165, 4.34170440711450924310729035425, 4.87101435940665416623687297247, 6.11641596010963479208704630285, 6.96493127503529042768200119618, 7.963740454189240180751469703319, 9.041000011219745566283182683749, 9.597942395510857276802733500541, 10.10899718311932251126679285485