L(s) = 1 | + (−0.654 + 0.755i)3-s + (0.309 − 2.14i)5-s + (1.22 + 2.68i)7-s + (−0.142 − 0.989i)9-s + (0.937 − 0.275i)11-s + (−0.279 + 0.611i)13-s + (1.42 + 1.64i)15-s + (2.67 − 1.72i)17-s + (1.94 + 1.25i)19-s + (−2.83 − 0.831i)21-s + (4.76 − 0.546i)23-s + (0.270 + 0.0795i)25-s + (0.841 + 0.540i)27-s + (0.860 − 0.552i)29-s + (2.27 + 2.62i)31-s + ⋯ |
L(s) = 1 | + (−0.378 + 0.436i)3-s + (0.138 − 0.961i)5-s + (0.463 + 1.01i)7-s + (−0.0474 − 0.329i)9-s + (0.282 − 0.0830i)11-s + (−0.0774 + 0.169i)13-s + (0.367 + 0.423i)15-s + (0.649 − 0.417i)17-s + (0.446 + 0.287i)19-s + (−0.617 − 0.181i)21-s + (0.993 − 0.114i)23-s + (0.0541 + 0.0159i)25-s + (0.161 + 0.104i)27-s + (0.159 − 0.102i)29-s + (0.408 + 0.471i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42018 + 0.200258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42018 + 0.200258i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-4.76 + 0.546i)T \) |
good | 5 | \( 1 + (-0.309 + 2.14i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.22 - 2.68i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.937 + 0.275i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.279 - 0.611i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.67 + 1.72i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.94 - 1.25i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.860 + 0.552i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.27 - 2.62i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.116 + 0.813i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.840 - 5.84i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (0.0622 - 0.0718i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + (0.944 + 2.06i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (2.17 - 4.76i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.64 - 1.89i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (11.2 + 3.30i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (0.504 + 0.148i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (7.15 + 4.59i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (5.51 - 12.0i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.46 + 10.1i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (0.750 - 0.866i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.23 + 8.60i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.88542074216828090711802922057, −9.811549327447127879442224648566, −9.023734671420894706217697549782, −8.478136566638809411981946964104, −7.24430049554683764211001783306, −5.91621762017920779578470045768, −5.22813088318247911902816952213, −4.44932908477892874849835586448, −2.93369343588132125503586288770, −1.26059292670423227345218248621,
1.15476916543169958494744063689, 2.76963036279105354709994429865, 4.00855745579209457978874101382, 5.23477840630220041709780025563, 6.34825546809860855036855494445, 7.19562993202427538352291402931, 7.71035125109279700824899089227, 8.991057003874320366553466881041, 10.24017877747070024850884666139, 10.68407469618146480820765904612