| L(s) = 1 | + (3.57 + 15.5i)2-s + (101. + 32.9i)3-s + (−230. + 111. i)4-s + (−155. − 605. i)5-s + (−151. + 1.70e3i)6-s + 328. i·7-s + (−2.56e3 − 3.19e3i)8-s + (3.90e3 + 2.83e3i)9-s + (8.88e3 − 4.58e3i)10-s + (1.27e4 + 1.76e4i)11-s + (−2.70e4 + 3.70e3i)12-s + (9.47e3 + 6.88e3i)13-s + (−5.11e3 + 1.17e3i)14-s + (4.21e3 − 6.65e4i)15-s + (4.07e4 − 5.13e4i)16-s + (4.21e4 + 1.29e5i)17-s + ⋯ |
| L(s) = 1 | + (0.223 + 0.974i)2-s + (1.25 + 0.407i)3-s + (−0.900 + 0.435i)4-s + (−0.248 − 0.968i)5-s + (−0.117 + 1.31i)6-s + 0.136i·7-s + (−0.625 − 0.780i)8-s + (0.594 + 0.432i)9-s + (0.888 − 0.458i)10-s + (0.873 + 1.20i)11-s + (−1.30 + 0.178i)12-s + (0.331 + 0.241i)13-s + (−0.133 + 0.0305i)14-s + (0.0833 − 1.31i)15-s + (0.621 − 0.783i)16-s + (0.505 + 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.177i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.211606 + 2.35896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.211606 + 2.35896i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.57 - 15.5i)T \) |
| 5 | \( 1 + (155. + 605. i)T \) |
| good | 3 | \( 1 + (-101. - 32.9i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 328. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.27e4 - 1.76e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-9.47e3 - 6.88e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-4.21e4 - 1.29e5i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (2.06e5 - 6.70e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (5.89e4 + 8.11e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (2.23e5 - 6.88e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (1.21e6 - 3.94e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-2.11e6 - 1.53e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (2.18e6 + 1.58e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 2.64e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-7.13e6 - 2.31e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (3.85e5 - 1.18e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (9.85e6 - 1.35e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.86e6 + 1.35e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (1.86e7 - 6.05e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-2.63e7 - 8.55e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.22e6 - 8.91e5i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (4.74e7 + 1.54e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (1.65e7 - 5.38e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (5.43e7 - 3.94e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-3.67e7 + 1.13e8i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.85587630022211812434284359801, −12.31326968558695353562897197547, −10.13739563559091931475109252564, −8.897005581328295674342927441158, −8.637821766116122366646287256596, −7.43402283127860453035427367625, −5.95695771208247228199432737968, −4.33015205650907030262540244805, −3.80457281504006381750435771876, −1.70707706873748243840638574767,
0.51820360860157611681448483021, 2.16300654714318978331602315876, 3.10163058726333426430677067700, 3.97040235374377083740123729444, 6.02742161906900527346905156005, 7.53408110400367392603117096792, 8.664502600391873183588807452030, 9.539090674729521084852950108086, 10.93196146989429965597734064139, 11.58768331696639633110683876496
