L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.68 + 1.46i)5-s + (−2.30 + 1.29i)7-s − 0.999·8-s + (2.11 − 0.728i)10-s + (−1.11 + 0.641i)11-s + 6.14·13-s + (−0.0298 + 2.64i)14-s + (−0.5 + 0.866i)16-s + (5.64 − 3.26i)17-s + (5.22 + 3.01i)19-s + (0.426 − 2.19i)20-s + 1.28i·22-s + (1.43 − 2.49i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.754 + 0.656i)5-s + (−0.871 + 0.490i)7-s − 0.353·8-s + (0.668 − 0.230i)10-s + (−0.335 + 0.193i)11-s + 1.70·13-s + (−0.00798 + 0.707i)14-s + (−0.125 + 0.216i)16-s + (1.37 − 0.790i)17-s + (1.19 + 0.692i)19-s + (0.0953 − 0.490i)20-s + 0.273i·22-s + (0.300 − 0.519i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91178 - 0.228710i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91178 - 0.228710i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-1.68 - 1.46i)T \) |
| 7 | \( 1 + (2.30 - 1.29i)T \) |
good | 11 | \( 1 + (1.11 - 0.641i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.14T + 13T^{2} \) |
| 17 | \( 1 + (-5.64 + 3.26i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.22 - 3.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.43 + 2.49i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.35iT - 29T^{2} \) |
| 31 | \( 1 + (7.49 - 4.32i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.25 - 4.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.71T + 41T^{2} \) |
| 43 | \( 1 - 5.35iT - 43T^{2} \) |
| 47 | \( 1 + (0.698 + 0.403i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.33 + 5.77i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.798 - 1.38i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.50 + 3.17i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.67 + 2.69i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 + (6.20 + 10.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.59 - 9.68i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.74iT - 83T^{2} \) |
| 89 | \( 1 + (1.81 - 3.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8.76T + 97T^{2} \) |
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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
−10.53197684013427816091073299101, −9.783288680583635891022552814549, −9.226431523938550201440736003074, −7.977949827836887669630744185827, −6.72358911551705334518682727072, −5.91700024385466538034672291144, −5.21792094017389870254540275335, −3.45420651139990836610158720984, −3.00343141268350191149152956167, −1.45928225515154012800531253047,
1.16738585874632197492703886812, 3.16767493328522118615791333009, 4.03362559821778314599795659918, 5.56022392645800507360564809554, 5.82382104507898618338704244928, 6.97699067855537122425789377852, 7.937748373829595466277107332299, 8.905982934941856488836312378947, 9.603834717457850857144730610716, 10.49942644415847569298788848310