| L(s) = 1 | + (−11.2 − 19.4i)2-s + (4 − 6.92i)4-s + (246. − 427. i)5-s + (381.5 + 660. i)7-s − 1.16e4·8-s − 1.10e4·10-s + (2.84e4 + 4.92e4i)11-s + (3.65e4 − 6.32e4i)13-s + (8.56e3 − 1.48e4i)14-s + (1.28e5 + 2.23e5i)16-s + 1.68e5·17-s − 5.98e5·19-s + (−1.97e3 − 3.42e3i)20-s + (6.38e5 − 1.10e6i)22-s + (−1.20e6 + 2.07e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.496 − 0.859i)2-s + (0.00781 − 0.0135i)4-s + (0.176 − 0.306i)5-s + (0.0600 + 0.104i)7-s − 1.00·8-s − 0.350·10-s + (0.585 + 1.01i)11-s + (0.354 − 0.614i)13-s + (0.0595 − 0.103i)14-s + (0.492 + 0.852i)16-s + 0.488·17-s − 1.05·19-s + (−0.00276 − 0.00478i)20-s + (0.581 − 1.00i)22-s + (−0.894 + 1.54i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.41024 - 0.248663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41024 - 0.248663i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (11.2 + 19.4i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (-246. + 427. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-381.5 - 660. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.84e4 - 4.92e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-3.65e4 + 6.32e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 - 1.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.98e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (1.20e6 - 2.07e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-2.32e6 - 4.03e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-9.12e5 + 1.57e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.42e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.49e7 - 2.58e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (3.87e6 + 6.71e6i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (1.54e7 + 2.68e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 - 9.90e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-6.26e7 + 1.08e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.63e7 - 1.32e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.60e8 + 2.77e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.09e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.64e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-5.76e7 - 9.99e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.07e8 - 5.32e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + 3.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-1.17e8 - 2.03e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−12.22064613347291396489640069762, −11.30978175983721714849694218741, −10.14393828833266518764824078151, −9.423126151297429827430649882872, −8.216317923931249081139654581912, −6.60339080968757616424987224872, −5.28381883090985909934251568145, −3.57630027374152330617356071093, −2.04184661868414367835218443316, −1.05841402512090494593719708471,
0.55760579713395727671144166622, 2.54930247512194649086234564507, 4.08409916440918738605021116212, 6.08706925241758656838181449289, 6.63547446718084765628982677954, 8.133863032976531301380379137263, 8.804389538192932872579541036140, 10.20494195402749378137549416892, 11.45401184559618849342034572478, 12.48910305881872107209700702478
