L(s) = 1 | + (−0.169 + 1.40i)2-s + (−1.94 − 0.477i)4-s + (−0.345 − 0.597i)5-s + (−2.63 − 0.222i)7-s + (0.999 − 2.64i)8-s + (0.897 − 0.382i)10-s + (1.63 − 2.82i)11-s + 5.27·13-s + (0.760 − 3.66i)14-s + (3.54 + 1.85i)16-s + (2.20 + 1.27i)17-s + (−0.484 + 0.279i)19-s + (0.385 + 1.32i)20-s + (3.68 + 2.76i)22-s + (2.50 − 1.44i)23-s + ⋯ |
L(s) = 1 | + (−0.120 + 0.992i)2-s + (−0.971 − 0.238i)4-s + (−0.154 − 0.267i)5-s + (−0.996 − 0.0840i)7-s + (0.353 − 0.935i)8-s + (0.283 − 0.121i)10-s + (0.491 − 0.851i)11-s + 1.46·13-s + (0.203 − 0.979i)14-s + (0.886 + 0.463i)16-s + (0.534 + 0.308i)17-s + (−0.111 + 0.0642i)19-s + (0.0861 + 0.296i)20-s + (0.786 + 0.590i)22-s + (0.522 − 0.301i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08585 + 0.111613i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08585 + 0.111613i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.169 - 1.40i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.63 + 0.222i)T \) |
good | 5 | \( 1 + (0.345 + 0.597i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.27T + 13T^{2} \) |
| 17 | \( 1 + (-2.20 - 1.27i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.484 - 0.279i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.50 + 1.44i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.444iT - 29T^{2} \) |
| 31 | \( 1 + (-4.45 + 7.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.00 - 3.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9.76iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (2.20 + 3.81i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.17 - 4.71i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (8.59 + 4.96i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.23 - 9.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.45 - 2.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.29iT - 71T^{2} \) |
| 73 | \( 1 + (5.28 + 3.05i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.01 - 2.89i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.83iT - 83T^{2} \) |
| 89 | \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.42iT - 97T^{2} \) |
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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
−10.70486710689050677179090741084, −9.923394781286195586667942034529, −8.737837623639323930895632948967, −8.526878728003689829105662952151, −7.18374772855856730624466088962, −6.26551600524358938421566384694, −5.72104049849678213089545731491, −4.22162790974360749136260610525, −3.37294011858378742114857462209, −0.823192453831704747953172461389,
1.34757179564338085050056674305, 3.01633659155385262486466392717, 3.72573403635759336872658722406, 4.99147494016450177703411566182, 6.28873478521919132403675187311, 7.28414958549030961868116671350, 8.556322387738904818975822301919, 9.276239461043339666424198688162, 10.07863270178289041261243877651, 10.87017822609972326311501948194