L(s) = 1 | + (1.30 − 2.26i)2-s + (−1.11 − 1.93i)3-s + (−2.42 − 4.20i)4-s + (1.11 − 1.93i)5-s − 5.85·6-s − 7.47·8-s + (−1 + 1.73i)9-s + (−2.92 − 5.06i)10-s + (1.5 + 2.59i)11-s + (−5.42 + 9.39i)12-s + 13-s − 5.00·15-s + (−4.92 + 8.53i)16-s + (0.736 + 1.27i)17-s + (2.61 + 4.53i)18-s + (1.5 − 2.59i)19-s + ⋯ |
L(s) = 1 | + (0.925 − 1.60i)2-s + (−0.645 − 1.11i)3-s + (−1.21 − 2.10i)4-s + (0.499 − 0.866i)5-s − 2.38·6-s − 2.64·8-s + (−0.333 + 0.577i)9-s + (−0.925 − 1.60i)10-s + (0.452 + 0.783i)11-s + (−1.56 + 2.71i)12-s + 0.277·13-s − 1.29·15-s + (−1.23 + 2.13i)16-s + (0.178 + 0.309i)17-s + (0.617 + 1.06i)18-s + (0.344 − 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06804 + 1.60564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06804 + 1.60564i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-1.30 + 2.26i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.11 + 1.93i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.11 + 1.93i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.736 - 1.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.11 + 7.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.35 - 4.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (3.73 - 6.47i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.73 - 6.47i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.736 + 1.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + (1.35 + 2.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.35 + 2.34i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.11 + 1.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−10.28591927399010808511086711188, −9.415776598239538169082884989601, −8.547058384835467774897959403444, −6.93076587463328469304648565271, −6.11586339675428518223589975536, −5.07530185526766253034894590276, −4.45700126473678639780881347558, −2.91831266741466623778743797806, −1.65101066267164557495071182343, −0.949291099823486747223309816262,
3.20905148485737465340353365424, 3.95607249836668106297901538249, 5.09806230723001197364171513873, 5.71177127466275601571271082886, 6.43656087010451697196656727330, 7.29659759714342890931595266705, 8.367781993969385686579002588395, 9.388029193672217040499140666298, 10.22058679508675968019203746393, 11.24787954536451516340043757789