L(s) = 1 | + (−0.551 − 0.955i)2-s + (0.391 − 0.678i)4-s + 0.105·5-s − 3.07·8-s + (−0.0581 − 0.100i)10-s − 3.33·11-s + (1.23 + 2.14i)13-s + (0.909 + 1.57i)16-s + (−0.806 − 1.39i)17-s + (−3.84 + 6.65i)19-s + (0.0413 − 0.0715i)20-s + (1.84 + 3.18i)22-s + 1.89·23-s − 4.98·25-s + (1.36 − 2.36i)26-s + ⋯ |
L(s) = 1 | + (−0.389 − 0.675i)2-s + (0.195 − 0.339i)4-s + 0.0471·5-s − 1.08·8-s + (−0.0183 − 0.0318i)10-s − 1.00·11-s + (0.343 + 0.595i)13-s + (0.227 + 0.393i)16-s + (−0.195 − 0.338i)17-s + (−0.881 + 1.52i)19-s + (0.00924 − 0.0160i)20-s + (0.392 + 0.679i)22-s + 0.395·23-s − 0.997·25-s + (0.268 − 0.464i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4059294604\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4059294604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.551 + 0.955i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 0.105T + 5T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 + (-1.23 - 2.14i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.806 + 1.39i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.84 - 6.65i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 + (4.64 - 8.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.63 + 8.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.991 + 1.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.74 - 6.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.77 - 6.53i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.59 - 2.76i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.98 + 8.64i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.22 - 3.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.83 + 4.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.98 - 8.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + (2.36 + 4.09i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.84 + 6.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.584 - 1.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.01 - 5.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.90 + 3.29i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.884930184058735543590033805639, −9.232739773252226083821471697738, −8.300499423685473518504675073510, −7.52008118242225364408671489202, −6.31620740735829287757456847999, −5.80899535305898113114334656317, −4.66849284595575029733714466805, −3.49619909267232752581310938667, −2.41281920301729260379776530964, −1.49970761956413755700382368555,
0.17894846591443560373161402463, 2.28347172434467710445462405380, 3.16216830123756714300905253351, 4.38081692401445873985806769445, 5.51058842118102229809312279943, 6.25825835496212433565456999493, 7.13517730727528744873470233389, 7.82739614439402438205404261747, 8.536936562867710984985274764754, 9.166769604766946116525273398250