L(s) = 1 | + (−0.228 + 0.395i)2-s + (−0.866 − 0.5i)3-s + (0.895 + 1.55i)4-s + (2.18 − 0.456i)5-s + (0.395 − 0.228i)6-s + (0.866 + 1.5i)7-s − 1.73·8-s + (−1 − 1.73i)9-s + (−0.319 + 0.970i)10-s + (−2.29 − 1.32i)11-s − 1.79i·12-s + (−3.46 − i)13-s − 0.791·14-s + (−2.12 − 0.698i)15-s + (−1.39 + 2.41i)16-s + (3.96 − 2.29i)17-s + ⋯ |
L(s) = 1 | + (−0.161 + 0.279i)2-s + (−0.499 − 0.288i)3-s + (0.447 + 0.775i)4-s + (0.978 − 0.204i)5-s + (0.161 − 0.0932i)6-s + (0.327 + 0.566i)7-s − 0.612·8-s + (−0.333 − 0.577i)9-s + (−0.100 + 0.306i)10-s + (−0.690 − 0.398i)11-s − 0.517i·12-s + (−0.960 − 0.277i)13-s − 0.211·14-s + (−0.548 − 0.180i)15-s + (−0.348 + 0.604i)16-s + (0.962 − 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.826184 + 0.193945i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.826184 + 0.193945i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.18 + 0.456i)T \) |
| 13 | \( 1 + (3.46 + i)T \) |
good | 2 | \( 1 + (0.228 - 0.395i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.866 - 1.5i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.29 + 1.32i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.96 + 2.29i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.96 + 2.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.29 + 3.96i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.20iT - 31T^{2} \) |
| 37 | \( 1 + (3.96 - 6.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.29 - 1.32i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.16 + 5.29i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 - 7.58iT - 53T^{2} \) |
| 59 | \( 1 + (12.0 - 6.97i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.708 - 1.22i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.504 + 0.873i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.08 + 3.51i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 + (-8.29 - 4.78i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.70 - 9.87i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−15.07538406314450356441436232835, −13.87482349473606592847938302402, −12.39557749092135279014793686982, −12.05466663240329074341459267750, −10.47381266725883982644422718256, −9.060735280943552566615311634674, −7.86497378911316177475538725302, −6.41429547456782242924466642019, −5.37045689866446928364770460358, −2.71184241147790884934468388573,
2.19809918440530383914969316923, 5.02036978095841196206089201887, 6.02045540426939672321457339109, 7.57355229296938211174650041154, 9.588076353283201091120387637768, 10.38062410762900683023618625008, 11.06326207813602762078210028112, 12.47675690648557051738466946161, 13.98233370935528704454391585536, 14.63585377966136100124005588017