L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.71 − 0.230i)3-s + (−0.499 + 0.866i)4-s + (−1.00 + 0.578i)5-s + (−1.05 − 1.37i)6-s + (0.296 + 2.62i)7-s + 0.999·8-s + (2.89 − 0.790i)9-s + (1.00 + 0.578i)10-s + (−1.85 + 3.21i)11-s + (−0.658 + 1.60i)12-s + (0.361 + 3.58i)13-s + (2.12 − 1.57i)14-s + (−1.58 + 1.22i)15-s + (−0.5 − 0.866i)16-s + (0.0128 − 0.0221i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.991 − 0.132i)3-s + (−0.249 + 0.433i)4-s + (−0.448 + 0.258i)5-s + (−0.431 − 0.559i)6-s + (0.111 + 0.993i)7-s + 0.353·8-s + (0.964 − 0.263i)9-s + (0.316 + 0.182i)10-s + (−0.559 + 0.969i)11-s + (−0.190 + 0.462i)12-s + (0.100 + 0.994i)13-s + (0.568 − 0.419i)14-s + (−0.409 + 0.315i)15-s + (−0.125 − 0.216i)16-s + (0.00310 − 0.00537i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42805 + 0.344038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42805 + 0.344038i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.71 + 0.230i)T \) |
| 7 | \( 1 + (-0.296 - 2.62i)T \) |
| 13 | \( 1 + (-0.361 - 3.58i)T \) |
good | 5 | \( 1 + (1.00 - 0.578i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.85 - 3.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.0128 + 0.0221i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.122 - 0.211i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.36 + 1.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.51iT - 29T^{2} \) |
| 31 | \( 1 + (-2.24 + 3.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.29 + 4.79i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.471iT - 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + (0.203 - 0.117i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.44 + 3.72i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.80 + 1.61i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.2 + 5.91i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.24 + 3.60i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + (-0.784 + 1.35i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.01 - 5.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.64iT - 83T^{2} \) |
| 89 | \( 1 + (8.11 - 4.68i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−10.90196114261762420901026698092, −9.657035441872187867586312963540, −9.294998313697630858439806942471, −8.276075922433014446957864341915, −7.58946975329063516329210572126, −6.61033208789187992255001622305, −4.95116795890337706862948733373, −3.89637063845067737285241889559, −2.71454484821187052712230096819, −1.85955046215319683431320676917,
0.909456548623422461847486382586, 2.92387174741680304687104322374, 4.01306579765940373694002892619, 5.03955654909343359827686614202, 6.35510057700756866951917549573, 7.54947968341762923976761393635, 8.018854425124146103021712888282, 8.664603777775978660784587933122, 9.822097344356546555689005834802, 10.44787604848212712838531586234