L(s) = 1 | + (−364.5 + 210. i)3-s + (−1.02e3 − 1.77e3i)4-s + (−3.85e4 − 2.21e4i)7-s + (8.85e4 − 1.53e5i)9-s + (7.46e5 + 4.30e5i)12-s + 1.22e6i·13-s + (−2.09e6 + 3.63e6i)16-s + (1.26e7 + 7.28e6i)19-s + (1.87e7 − 5.63e4i)21-s + (2.44e7 + 4.22e7i)25-s + 7.45e7i·27-s + (2.74e5 + 9.10e7i)28-s + (1.01e8 − 5.85e7i)31-s − 3.62e8·36-s + (3.31e8 − 5.74e8i)37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.867 − 0.497i)7-s + (0.5 − 0.866i)9-s + (0.866 + 0.499i)12-s + 0.911i·13-s + (−0.499 + 0.866i)16-s + (1.16 + 0.675i)19-s + (0.999 − 0.00301i)21-s + (0.499 + 0.866i)25-s + 0.999i·27-s + (0.00301 + 0.999i)28-s + (0.636 − 0.367i)31-s − 36-s + (0.786 − 1.36i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.708006 + 0.352308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.708006 + 0.352308i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (364.5 - 210. i)T \) |
| 7 | \( 1 + (3.85e4 + 2.21e4i)T \) |
good | 2 | \( 1 + (1.02e3 + 1.77e3i)T^{2} \) |
| 5 | \( 1 + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.22e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.26e7 - 7.28e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.01e8 + 5.85e7i)T + (1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-3.31e8 + 5.74e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.76e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.07e10 - 6.21e9i)T + (2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.07e10 - 1.85e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.31e20T^{2} \) |
| 73 | \( 1 + (-2.13e9 + 1.23e9i)T + (1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.07e10 - 1.85e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.26e11iT - 7.15e21T^{2} \) |
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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
−15.87120685711821636619520165145, −14.48745541171504812652862850540, −13.14209106355966646318866666612, −11.52488076636173629532336542752, −10.17615987994714299937514811430, −9.354901834655399639726450017452, −6.76196309098378319371975946692, −5.44999995668248908107810621960, −3.95667666988824479989651290073, −0.984079518132159641243900442276,
0.47020608590198100993697020433, 2.99152536230134208411209198025, 5.05403879296783702763291689479, 6.69072964154620511027888855633, 8.183724634892179577670966637019, 9.886125155080246848289007398095, 11.68185929555701624873536147365, 12.66519505913501434828279708948, 13.54625619543939111656750465412, 15.70239018114830833869061258745