L(s) = 1 | + 3.20i·3-s − 2·4-s + 0.837i·5-s + 2.87i·7-s − 7.25·9-s − 6.13i·11-s − 6.40i·12-s − 4.57·13-s − 2.68·15-s + 4·16-s + (−1.31 + 3.90i)17-s + 2·19-s − 1.67i·20-s − 9.21·21-s + 5.48i·23-s + ⋯ |
L(s) = 1 | + 1.84i·3-s − 4-s + 0.374i·5-s + 1.08i·7-s − 2.41·9-s − 1.85i·11-s − 1.84i·12-s − 1.26·13-s − 0.692·15-s + 16-s + (−0.320 + 0.947i)17-s + 0.458·19-s − 0.374i·20-s − 2.00·21-s + 1.14i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.236888 - 0.330082i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.236888 - 0.330082i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (1.31 - 3.90i)T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 - 3.20iT - 3T^{2} \) |
| 5 | \( 1 - 0.837iT - 5T^{2} \) |
| 7 | \( 1 - 2.87iT - 7T^{2} \) |
| 11 | \( 1 + 6.13iT - 11T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 5.48iT - 23T^{2} \) |
| 29 | \( 1 + 1.67iT - 29T^{2} \) |
| 31 | \( 1 + 1.40iT - 31T^{2} \) |
| 37 | \( 1 + 3.85iT - 37T^{2} \) |
| 41 | \( 1 - 5.11iT - 41T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 + 9.90iT - 61T^{2} \) |
| 67 | \( 1 + 8.73T + 67T^{2} \) |
| 71 | \( 1 + 8.76iT - 71T^{2} \) |
| 73 | \( 1 - 15.6iT - 73T^{2} \) |
| 79 | \( 1 - 8.56iT - 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 3.05T + 89T^{2} \) |
| 97 | \( 1 + 7.16iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.90572667224160288491426696172, −9.898728039645114916239344838174, −9.389710641114898435228870660769, −8.691430590675312311093511296991, −8.052625779825755521412080100449, −6.04664006006771936282346847293, −5.44523419300502771090311120182, −4.69513562021482626557725425641, −3.56328137257221388213857688324, −2.92865930797399720515730928875,
0.22131400347888335048552973460, 1.45371145472481927430521357821, 2.77110844290525030950734551725, 4.54627747059582986621185878634, 5.07070403292974928962161616436, 6.64634383287841302336075498837, 7.28103814390627964068188713534, 7.74542922762109005204407670309, 8.827021303044983350686348556274, 9.664499992912570769560882735074