L(s) = 1 | + (0.5 + 0.866i)3-s + (−1.5 + 2.59i)5-s + (−2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s − 4·13-s − 3·15-s + (−2 + 3.46i)19-s + (−0.500 − 2.59i)21-s + (−2 − 3.46i)25-s − 0.999·27-s + 9·29-s + (−0.5 − 0.866i)31-s + (−1.5 + 2.59i)33-s + (6 − 5.19i)35-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.670 + 1.16i)5-s + (−0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s − 1.10·13-s − 0.774·15-s + (−0.458 + 0.794i)19-s + (−0.109 − 0.566i)21-s + (−0.400 − 0.692i)25-s − 0.192·27-s + 1.67·29-s + (−0.0898 − 0.155i)31-s + (−0.261 + 0.452i)33-s + (1.01 − 0.878i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.342221 + 0.816595i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.342221 + 0.816595i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−11.94579060733661460199508869257, −10.72212498015924464697900610613, −10.11889823429807145466715362340, −9.373484429776761323051935071549, −7.989817115489693797396006955504, −7.09738994596427036984048123802, −6.35255528811349985454977247266, −4.63822389211924266936234799740, −3.62149717635485139103694469786, −2.62065621479112232367859102735,
0.58902855732605520581865986559, 2.63941880620175372678622032611, 3.99645516252522755732828313743, 5.20838068802029401398595083824, 6.44685836039627498208029868992, 7.42436504408232691378467766690, 8.631449566426585166342067617442, 8.984289366136348496583847792823, 10.15020335236476176728793008828, 11.53027734566158680098335300150