L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)11-s + 6·13-s + 0.999·15-s + (−2.5 − 4.33i)17-s + (0.5 − 0.866i)19-s + (3.5 − 6.06i)23-s + (2 + 3.46i)25-s − 5·27-s + 2·29-s + (−2.5 − 4.33i)31-s + (−1.5 + 2.59i)33-s + (−1.5 + 2.59i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.333 − 0.577i)9-s + (−0.452 − 0.783i)11-s + 1.66·13-s + 0.258·15-s + (−0.606 − 1.05i)17-s + (0.114 − 0.198i)19-s + (0.729 − 1.26i)23-s + (0.400 + 0.692i)25-s − 0.962·27-s + 0.371·29-s + (−0.449 − 0.777i)31-s + (−0.261 + 0.452i)33-s + (−0.246 + 0.427i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.993662 - 0.660966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.993662 - 0.660966i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-2.5 + 4.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.5 - 12.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−11.21484936234559258146857679980, −10.48379510859631582914119949999, −9.140068152129027869207008089391, −8.425911655133464017826388398787, −7.16193884214576760508688089417, −6.51651468029959238085095983353, −5.49919240724276569573065721133, −4.02234528839430831765227611664, −2.85402239447732805081880571762, −0.908537998703006635301490806447,
1.69223585932844597424257716771, 3.60519013220142018833328961052, 4.59243820186855917607555370604, 5.53509259177546425069227481568, 6.71241178640570830388655050998, 7.901803104182270669397853063498, 8.694661047628741924389771571233, 9.752489567418408370724052408402, 10.73907442309271620243213136897, 11.15145070133530849482853973007