L(s) = 1 | + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (0.207 + 0.978i)5-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.5 + 0.866i)10-s + (0.104 − 0.994i)14-s + (−0.978 + 0.207i)16-s + (0.978 − 0.207i)17-s + (0.994 + 0.104i)19-s + (−0.951 + 0.309i)20-s + 23-s + (−0.913 + 0.406i)25-s + (0.743 − 0.669i)28-s + (0.913 + 0.406i)29-s + ⋯ |
L(s) = 1 | + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (0.207 + 0.978i)5-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.5 + 0.866i)10-s + (0.104 − 0.994i)14-s + (−0.978 + 0.207i)16-s + (0.978 − 0.207i)17-s + (0.994 + 0.104i)19-s + (−0.951 + 0.309i)20-s + 23-s + (−0.913 + 0.406i)25-s + (0.743 − 0.669i)28-s + (0.913 + 0.406i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9607830447 + 3.165131992i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9607830447 + 3.165131992i\) |
\(L(1)\) |
\(\approx\) |
\(1.303436192 + 0.9883013513i\) |
\(L(1)\) |
\(\approx\) |
\(1.303436192 + 0.9883013513i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.743 + 0.669i)T \) |
| 5 | \( 1 + (0.207 + 0.978i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.978 - 0.207i)T \) |
| 19 | \( 1 + (0.994 + 0.104i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.913 + 0.406i)T \) |
| 31 | \( 1 + (0.743 + 0.669i)T \) |
| 37 | \( 1 + (-0.994 + 0.104i)T \) |
| 41 | \( 1 + (0.587 - 0.809i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.406 + 0.913i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.994 - 0.104i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.207 + 0.978i)T \) |
| 73 | \( 1 + (-0.587 - 0.809i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.743 + 0.669i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.74808411479777672548298243322, −19.79669328448609838976210315153, −19.19507738156546407168602545329, −18.49612264617229013999412237998, −17.495292938628573493441304490492, −16.51711867427811795071110150077, −15.76397932144397096110579234524, −15.174485623981622467193463685080, −14.09132285904093491486531883862, −13.46770774495543011263043922424, −12.60333557033686176021558139030, −12.18629236488115258350494700148, −11.46321937703039025266676495931, −10.30030375273024880224429516450, −9.56223732887614488389587045354, −9.01565416696037191462219953721, −7.94088105397735872583197254077, −6.65750446007801851816691508897, −5.74209216009934192633439602199, −5.24835837453130509201964242224, −4.33888718426883482486961031543, −3.27670978697529581739283813517, −2.52628579938571604283813985328, −1.38198529071448683536029014741, −0.55796604963211883318070745341,
1.06233923140704293018332704927, 2.72937302037187120390779262782, 3.2253622382586021996227043715, 4.07236800707092282838981129139, 5.19802666022111852257402192361, 5.96928823024977352639647715798, 7.05658741623343903334771803229, 7.13398075791364893493347278373, 8.236676308906188257054988753172, 9.40406923697097885696870154522, 10.25646546303130951692157214271, 11.03930571268653979739850211602, 12.00634262374384411699007375888, 12.75328115218672320648549164299, 13.815218773356434575517488956460, 14.04269148402019723099037332691, 14.854110623136744579162484197685, 15.85390567549354695670659451712, 16.27035073796950375252163737991, 17.36201478791140975793511552894, 17.75276668640009044087869735462, 18.89213249651051096739313911777, 19.51735194741150749204493310153, 20.72913514147178505975193953118, 21.12072196163265208061489282984