| L(s) = 1 | + (−1.03 − 0.965i)2-s + (0.539 + 1.64i)3-s + (0.135 + 1.99i)4-s + (−1.86 − 1.23i)5-s + (1.03 − 2.22i)6-s − 3.63i·7-s + (1.78 − 2.19i)8-s + (−2.41 + 1.77i)9-s + (0.734 + 3.07i)10-s + (2.73 − 1.98i)11-s + (−3.21 + 1.29i)12-s + (−1.11 − 0.809i)13-s + (−3.51 + 3.75i)14-s + (1.02 − 3.73i)15-s + (−3.96 + 0.539i)16-s + (5.90 − 1.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.730 − 0.682i)2-s + (0.311 + 0.950i)3-s + (0.0676 + 0.997i)4-s + (−0.833 − 0.552i)5-s + (0.421 − 0.906i)6-s − 1.37i·7-s + (0.631 − 0.775i)8-s + (−0.806 + 0.591i)9-s + (0.232 + 0.972i)10-s + (0.825 − 0.599i)11-s + (−0.927 + 0.374i)12-s + (−0.309 − 0.224i)13-s + (−0.938 + 1.00i)14-s + (0.264 − 0.964i)15-s + (−0.990 + 0.134i)16-s + (1.43 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.664359 - 0.498753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.664359 - 0.498753i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.03 + 0.965i)T \) |
| 3 | \( 1 + (-0.539 - 1.64i)T \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| good | 7 | \( 1 + 3.63iT - 7T^{2} \) |
| 11 | \( 1 + (-2.73 + 1.98i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.11 + 0.809i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.90 + 1.91i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.04 + 1.31i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.33 + 3.15i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (7.29 + 2.37i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.22 - 0.721i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.973 + 0.707i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.55 - 3.51i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 7.53iT - 43T^{2} \) |
| 47 | \( 1 + (-1.07 + 3.29i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (6.24 + 2.02i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.35 - 3.89i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.39 - 5.37i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.34 - 1.08i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.0828 - 0.254i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.70 + 3.41i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-13.3 - 4.35i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.28 - 13.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.544 + 0.750i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.81 + 8.66i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.33147286000800705266832660909, −10.60175198894882376471608612818, −9.666866367442928638374235397345, −8.945399169320295635502543412383, −7.88704341172970313971547645258, −7.22129952932316803611798166917, −5.07132276650460044522800182575, −3.88311106270136489412669739061, −3.31578728546520292769210229617, −0.823587150600133424289411502638,
1.67308416474127208453741125070, 3.25847002561468124795516477495, 5.33674268139835971765394128096, 6.31158873071779593341736110551, 7.34896288846342548660480781468, 7.87008694922512083826324790654, 8.992472593982153831524343295758, 9.619283778315364005204153505428, 11.17384744016115660002470032155, 11.91611643764836821765562120248
