| L(s) = 1 | + (0.987 + 0.156i)2-s + (0.0565 + 1.73i)3-s + (0.951 + 0.309i)4-s + (1.48 − 1.66i)5-s + (−0.214 + 1.71i)6-s + (−1.08 + 1.08i)7-s + (0.891 + 0.453i)8-s + (−2.99 + 0.195i)9-s + (1.73 − 1.41i)10-s + (−1.61 − 2.21i)11-s + (−0.481 + 1.66i)12-s + (0.355 + 2.24i)13-s + (−1.24 + 0.903i)14-s + (2.97 + 2.48i)15-s + (0.809 + 0.587i)16-s + (2.77 − 5.44i)17-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (0.0326 + 0.999i)3-s + (0.475 + 0.154i)4-s + (0.665 − 0.746i)5-s + (−0.0877 + 0.701i)6-s + (−0.410 + 0.410i)7-s + (0.315 + 0.160i)8-s + (−0.997 + 0.0653i)9-s + (0.547 − 0.447i)10-s + (−0.486 − 0.669i)11-s + (−0.138 + 0.480i)12-s + (0.0985 + 0.622i)13-s + (−0.332 + 0.241i)14-s + (0.767 + 0.641i)15-s + (0.202 + 0.146i)16-s + (0.673 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53878 + 0.599636i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53878 + 0.599636i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.987 - 0.156i)T \) |
| 3 | \( 1 + (-0.0565 - 1.73i)T \) |
| 5 | \( 1 + (-1.48 + 1.66i)T \) |
| good | 7 | \( 1 + (1.08 - 1.08i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.61 + 2.21i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.355 - 2.24i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.77 + 5.44i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (4.05 - 1.31i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.805 - 5.08i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-2.37 + 7.29i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.85 + 5.71i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.29 - 0.521i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (5.36 - 7.38i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-6.65 - 6.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.15 + 1.09i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.199 - 0.391i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.86 - 4.26i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.14 - 5.91i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.28 - 2.69i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (5.89 + 1.91i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.2 - 1.62i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-6.06 - 1.97i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (6.92 + 3.52i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (8.12 - 5.90i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.51 - 10.8i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−13.39778635094597024147432920806, −12.13633407028448576477755569168, −11.27105844921607857728026243167, −9.949824916641642350824766350524, −9.234216240543220211262076510219, −8.048579789289619341350114297945, −6.13877060818899774196333685312, −5.39305553813788925865544234080, −4.23157133730680949644448647605, −2.71175555699623959022338551693,
2.04730711244208248663716876959, 3.39690811089426487732510336894, 5.39034717897037561498767180784, 6.48893133163980359591370915373, 7.19895136074875159110147000308, 8.550246524411396943313348505523, 10.36032668383931714698006270223, 10.72933474886312101053405921466, 12.54203760721863677547183627306, 12.66623431900273693574894987718
