L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)11-s − 4·13-s + 0.999·15-s + (2 + 3.46i)19-s + (2.5 + 0.866i)21-s + (4 + 6.92i)23-s + (2 − 3.46i)25-s + 0.999·27-s + 3·29-s + (−2.5 + 4.33i)31-s + (−1.5 − 2.59i)33-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.452 + 0.783i)11-s − 1.10·13-s + 0.258·15-s + (0.458 + 0.794i)19-s + (0.545 + 0.188i)21-s + (0.834 + 1.44i)23-s + (0.400 − 0.692i)25-s + 0.192·27-s + 0.557·29-s + (−0.449 + 0.777i)31-s + (−0.261 − 0.452i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.067770453\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.067770453\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 2.59i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 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 | \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−9.781902136945554599033536265168, −9.241997096969628710423291823930, −7.936635458110483288090868889054, −7.44058839681814738828352544225, −6.55546476282777777743901221917, −5.32762618885712249627638982469, −4.70795440166699534158808531588, −3.88421380091931557956845279150, −2.75192264133369348604027484761, −1.09720465747572973934906707699,
0.53715370496901960073564035739, 2.45631239898981412053681327808, 2.89798763996411811623349970824, 4.46708519569930370277086110063, 5.44830099641157019875692403804, 6.07537297320229014400431827855, 7.13780004140830061247801903263, 7.62331958141968081156602846742, 8.803935598067136771225591031170, 9.212288569156205225775497691287