L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s − 1.41·5-s + (0.707 + 0.707i)6-s − i·8-s − 1.00i·9-s − 1.41i·10-s + (−0.707 + 0.707i)12-s − 1.41i·13-s + (−1.00 + 1.00i)15-s + 16-s + 1.00·18-s − 1.41i·19-s + 1.41·20-s + ⋯ |
L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s − 1.41·5-s + (0.707 + 0.707i)6-s − i·8-s − 1.00i·9-s − 1.41i·10-s + (−0.707 + 0.707i)12-s − 1.41i·13-s + (−1.00 + 1.00i)15-s + 16-s + 1.00·18-s − 1.41i·19-s + 1.41·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7932126558\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7932126558\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.496829874429999186713176280679, −8.607499437543141806002763592793, −8.143201281429004746227442363266, −7.40876616553390710291229569198, −6.93606943901318308781374465777, −5.81698232931991341548527294273, −4.71270027386828520876878175040, −3.73535947127141583352604707135, −2.89743417717597250372062452539, −0.67791867429010710534282028460,
1.82487088156763979650989484777, 3.09110159398979902564272386716, 4.02329185792607415393981072185, 4.26706630360236553004198159315, 5.45128311130258175308186879408, 7.04160093884239663316382784670, 8.078032511902532995696098990513, 8.459772297917440529955346282450, 9.395024689720460936250570184979, 10.03180080703890487845072711645