L(s) = 1 | + (0.289 + 0.502i)2-s + (0.946 + 1.63i)3-s + (0.831 − 1.44i)4-s + 1.47·5-s + (−0.548 + 0.950i)6-s + 2.12·8-s + (−0.289 + 0.502i)9-s + (0.427 + 0.739i)10-s + (−0.289 − 0.502i)11-s + 3.14·12-s + (−0.128 + 3.60i)13-s + (1.39 + 2.41i)15-s + (−1.04 − 1.81i)16-s + (0.598 − 1.03i)17-s − 0.336·18-s + (0.230 − 0.399i)19-s + ⋯ |
L(s) = 1 | + (0.204 + 0.355i)2-s + (0.546 + 0.946i)3-s + (0.415 − 0.720i)4-s + 0.659·5-s + (−0.223 + 0.387i)6-s + 0.751·8-s + (−0.0966 + 0.167i)9-s + (0.135 + 0.233i)10-s + (−0.0874 − 0.151i)11-s + 0.908·12-s + (−0.0357 + 0.999i)13-s + (0.359 + 0.623i)15-s + (−0.261 − 0.453i)16-s + (0.145 − 0.251i)17-s − 0.0792·18-s + (0.0528 − 0.0915i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.31562 + 0.919598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.31562 + 0.919598i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.128 - 3.60i)T \) |
good | 2 | \( 1 + (-0.289 - 0.502i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.946 - 1.63i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + (0.289 + 0.502i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.598 + 1.03i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.230 + 0.399i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.18 + 2.05i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.44 - 5.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 + (4.58 + 7.93i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.00 - 3.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.02 - 6.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 + (0.120 - 0.208i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.86 - 6.69i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.724 - 1.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.25 + 10.8i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + (1.24 + 2.15i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.82 - 13.5i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.48269374612427527758620041950, −9.724018601394590073266889043170, −9.273182458155870853294894801469, −8.161876495958905586269346762800, −6.87152506967442415187739369277, −6.21804703500295942248998175848, −5.09382296465203305565283742638, −4.34880512575019808267548990958, −2.98727383168121233517606937412, −1.66158221927596276142838311504,
1.59058216955735166432475461521, 2.50572230772502125393385968961, 3.45227645664299585687775169474, 4.88202757789733824037848100683, 6.17255330241092875729253294727, 7.01649251790707416892268119677, 8.048024213274473977572146526860, 8.248361771669931834203090836789, 9.782022238773617990460042799171, 10.43483234266453222337866063878