L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s + (0.898 − 2.04i)5-s − 6-s + (2.04 + 0.664i)7-s + (−0.951 + 0.309i)8-s + (−0.309 − 0.951i)9-s + (2.18 − 0.476i)10-s + (1.16 − 3.58i)11-s + (−0.587 − 0.809i)12-s + (−1.53 + 2.11i)13-s + (0.664 + 2.04i)14-s + (1.12 + 1.93i)15-s + (−0.809 − 0.587i)16-s + (5.98 − 1.94i)17-s + ⋯ |
L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.339 + 0.467i)3-s + (−0.154 + 0.475i)4-s + (0.401 − 0.915i)5-s − 0.408·6-s + (0.772 + 0.250i)7-s + (−0.336 + 0.109i)8-s + (−0.103 − 0.317i)9-s + (0.690 − 0.150i)10-s + (0.350 − 1.08i)11-s + (−0.169 − 0.233i)12-s + (−0.426 + 0.587i)13-s + (0.177 + 0.546i)14-s + (0.291 + 0.498i)15-s + (−0.202 − 0.146i)16-s + (1.45 − 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98160 + 0.573581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98160 + 0.573581i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (-0.898 + 2.04i)T \) |
| 31 | \( 1 + (-5.56 + 0.110i)T \) |
good | 7 | \( 1 + (-2.04 - 0.664i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-1.16 + 3.58i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.53 - 2.11i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.98 + 1.94i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.19 + 1.59i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (4.25 - 1.38i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.11 + 4.44i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 - 5.91iT - 37T^{2} \) |
| 41 | \( 1 + (-5.93 + 4.30i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.81 - 6.62i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (6.25 - 8.61i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.02 - 0.656i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.25 - 3.82i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 8.58iT - 67T^{2} \) |
| 71 | \( 1 + (-3.11 - 9.60i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.73 - 0.563i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.417 - 1.28i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.19 - 4.40i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.680 - 2.09i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.25 - 2.68i)T + (78.4 + 57.0i)T^{2} \) |
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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
−9.854029790037734343466189647033, −9.359284085289304115006294728851, −8.304264246066918231336147983464, −7.83304366013584469827287102468, −6.37848300300980847249928294032, −5.71797390357208386499692529449, −4.92490477709482915140223969819, −4.25209740776159236848851402332, −2.89466453202499879395459319357, −1.10685797824317394393171414975,
1.33508848537826753119815615254, 2.35899352380297199709196602065, 3.51737185609202302629968068531, 4.71661121749525448825006575480, 5.60080292412014550640056035077, 6.47069683989203382242247585440, 7.44504830160353965555526832468, 8.054985536465462929336827153051, 9.547027575959708091501962663132, 10.29304056668055175173290285990