L(s) = 1 | + (0.5 + 0.866i)7-s − 2·13-s + (1.5 + 0.866i)19-s + (0.5 + 0.866i)25-s + (1.5 − 0.866i)31-s + (−1 + 1.73i)37-s + 1.73i·43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)61-s + (−0.5 − 0.866i)73-s + (−1 − 1.73i)91-s + 97-s + (−0.5 − 0.866i)109-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)7-s − 2·13-s + (1.5 + 0.866i)19-s + (0.5 + 0.866i)25-s + (1.5 − 0.866i)31-s + (−1 + 1.73i)37-s + 1.73i·43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)61-s + (−0.5 − 0.866i)73-s + (−1 − 1.73i)91-s + 97-s + (−0.5 − 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.170760078\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.170760078\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 2T + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - 1.73iT - T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−9.146693244305479073729174041854, −8.098636429704326038154948349676, −7.71639180747147600790581683550, −6.83733614216917915903911867863, −5.91158456864191027091344266199, −5.04013905747245170667003445527, −4.70943401038473946306786990320, −3.24928138171835903276612853841, −2.56752116197349661251024892676, −1.45063366023305627670041675194,
0.75961566223952875040576801058, 2.19324552249228171479764119394, 3.06206064419376458676994437108, 4.20143540156681904659524904468, 4.92993896973130290987753981774, 5.46284629901417818555903742545, 6.96206432296125985913692154291, 7.09409056540165933756285349416, 7.899457029760147055175703262582, 8.777566555501710606406243005225