| L(s) = 1 | + (0.910 + 1.08i)2-s + (−0.881 + 0.471i)3-s + (−0.342 + 1.97i)4-s + (2.63 + 3.21i)5-s + (−1.31 − 0.525i)6-s + (1.69 − 1.13i)7-s + (−2.44 + 1.42i)8-s + (0.555 − 0.831i)9-s + (−1.07 + 5.77i)10-s + (4.28 − 1.29i)11-s + (−0.627 − 1.89i)12-s + (−2.85 − 2.34i)13-s + (2.76 + 0.802i)14-s + (−3.83 − 1.58i)15-s + (−3.76 − 1.34i)16-s + (−5.40 + 2.23i)17-s + ⋯ |
| L(s) = 1 | + (0.643 + 0.765i)2-s + (−0.509 + 0.272i)3-s + (−0.171 + 0.985i)4-s + (1.17 + 1.43i)5-s + (−0.536 − 0.214i)6-s + (0.640 − 0.427i)7-s + (−0.864 + 0.503i)8-s + (0.185 − 0.277i)9-s + (−0.340 + 1.82i)10-s + (1.29 − 0.391i)11-s + (−0.181 − 0.548i)12-s + (−0.791 − 0.649i)13-s + (0.739 + 0.214i)14-s + (−0.991 − 0.410i)15-s + (−0.941 − 0.337i)16-s + (−1.31 + 0.542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.972837 + 1.69238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.972837 + 1.69238i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.910 - 1.08i)T \) |
| 3 | \( 1 + (0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (-2.63 - 3.21i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.69 + 1.13i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.28 + 1.29i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (2.85 + 2.34i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (5.40 - 2.23i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.269 + 2.74i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-3.31 + 0.659i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.408 + 1.34i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (4.32 + 4.32i)T + 31iT^{2} \) |
| 37 | \( 1 + (-8.05 - 0.793i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.11 + 5.58i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (3.16 + 1.69i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-0.865 - 2.08i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.581 - 1.91i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-9.18 + 7.54i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (5.57 + 10.4i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-3.26 - 6.10i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (4.41 + 6.60i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.61 - 3.74i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (4.82 - 11.6i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (5.18 - 0.511i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.64 - 1.12i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-8.36 - 8.36i)T + 97iT^{2} \) |
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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
−11.29856527952223341466220108283, −11.07188061085043582539135923782, −9.839688343557355625622235018927, −8.924298017974186579425750759098, −7.46162847279033722018728702857, −6.65117474071293987463280314905, −6.07869801645399933885471275502, −4.96416030671723629055708461062, −3.77679122755935995462869579573, −2.40186815213619841114783573546,
1.34892526174985256318494801741, 2.14779708120168522583854824745, 4.43451595869398519607649070760, 4.93814299591513627581680717100, 5.90590997284402341621424262391, 6.85833707288566581119705862209, 8.790286086978980888398461498567, 9.255305605424715673570760741511, 10.11281113190142672854918111046, 11.46003739889699939334072327676
