L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 3·5-s + (0.499 + 0.866i)6-s + (1.5 + 2.59i)7-s + 0.999·8-s + (1 + 1.73i)9-s + (1.5 − 2.59i)10-s − 0.999·12-s − 3·14-s + (−1.5 + 2.59i)15-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s − 2·18-s + (3 + 5.19i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 1.34·5-s + (0.204 + 0.353i)6-s + (0.566 + 0.981i)7-s + 0.353·8-s + (0.333 + 0.577i)9-s + (0.474 − 0.821i)10-s − 0.288·12-s − 0.801·14-s + (−0.387 + 0.670i)15-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s − 0.471·18-s + (0.688 + 1.19i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.656654 + 0.665129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.656654 + 0.665129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.5 + 12.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6 - 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.88862639897481546215898488931, −10.92893618516931274626814179970, −9.759049823529102758096620835210, −8.538953124259253901670515954284, −7.88424527804085826723169300360, −7.48269930000821693675685326391, −6.02401962466572942600342984931, −4.94387245343016574498677669573, −3.61210072037522835804654178372, −1.75379147406999824219161000448,
0.77687803298958992433426839852, 3.05988982348991942765292979570, 4.07688747794639576191864552207, 4.72106010857320208054692707163, 6.89515850325704627624603732733, 7.68222143327719995431361848210, 8.536302744487944474932573638219, 9.571025502658160065546788980206, 10.43545904814204633665244208173, 11.33627212417940197412624706607