L(s) = 1 | + (1.41 + 2.44i)3-s + (1.81 + 1.04i)7-s + (−2.49 + 4.32i)9-s + (1.5 − 0.866i)11-s + (3.59 − 0.331i)13-s + (1.81 − 3.14i)17-s + (0.926 + 0.534i)19-s + 5.92i·21-s + (3.90 + 6.77i)23-s − 5.62·27-s + (0.263 + 0.456i)29-s − 5.84i·31-s + (4.24 + 2.44i)33-s + (−8.44 + 4.87i)37-s + (5.88 + 8.32i)39-s + ⋯ |
L(s) = 1 | + (0.816 + 1.41i)3-s + (0.685 + 0.395i)7-s + (−0.831 + 1.44i)9-s + (0.452 − 0.261i)11-s + (0.995 − 0.0918i)13-s + (0.439 − 0.762i)17-s + (0.212 + 0.122i)19-s + 1.29i·21-s + (0.815 + 1.41i)23-s − 1.08·27-s + (0.0489 + 0.0847i)29-s − 1.04i·31-s + (0.738 + 0.426i)33-s + (−1.38 + 0.801i)37-s + (0.942 + 1.33i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.530271915\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.530271915\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.59 + 0.331i)T \) |
good | 3 | \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.81 - 1.04i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.81 + 3.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.926 - 0.534i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.90 - 6.77i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.263 - 0.456i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.84iT - 31T^{2} \) |
| 37 | \( 1 + (8.44 - 4.87i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.69 - 2.13i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.67 - 8.09i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + (1.21 + 0.701i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.55 + 9.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.38 + 5.41i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (12.2 + 7.08i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 2.64iT - 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 15.7iT - 83T^{2} \) |
| 89 | \( 1 + (4.78 - 2.76i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.1 - 7.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.619557930705096063276900335749, −9.218504429947896448690505571536, −8.371647218240285722734297939702, −7.82161368708201884769249884485, −6.49152148289764570428868587687, −5.30756694592498334338400895862, −4.80228336511858842578402483564, −3.58608494490015767920456394889, −3.16207095982930803001104818946, −1.62668990578418939230222804524,
1.11684143707481314357475385776, 1.84564271996858583605448814497, 3.08689495825043471410033749603, 4.06103505812584032510054882119, 5.32382278131493428586176481831, 6.54672813337733903886661934985, 6.94768332050399799914564228088, 7.88180721074273551709123697845, 8.539365602241681585453061221485, 8.985005732669105115833726395340