| L(s) = 1 | + (−2.96e3 − 5.14e3i)2-s + (−2.39e6 + 4.14e6i)3-s + (2.50e8 − 4.34e8i)4-s + (3.53e9 + 6.12e9i)5-s + 2.84e10·6-s + (−1.40e12 + 1.11e12i)7-s + (−6.16e12 − 0.000244i)8-s + (−1.14e13 − 1.98e13i)9-s + (2.09e13 − 3.63e13i)10-s + (1.86e14 − 3.22e14i)11-s + (1.19e15 + 2.07e15i)12-s − 9.69e15·13-s + (9.90e15 + 3.92e15i)14-s − 3.38e16·15-s + (−1.16e17 − 2.01e17i)16-s + (−3.20e17 + 5.55e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.128 − 0.221i)2-s + (−0.288 + 0.499i)3-s + (0.467 − 0.809i)4-s + (0.258 + 0.448i)5-s + 0.147·6-s + (−0.783 + 0.620i)7-s − 0.495·8-s + (−0.166 − 0.288i)9-s + (0.0663 − 0.114i)10-s + (0.147 − 0.255i)11-s + (0.269 + 0.467i)12-s − 0.682·13-s + (0.238 + 0.0944i)14-s − 0.299·15-s + (−0.403 − 0.699i)16-s + (−0.461 + 0.799i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0826i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.387896279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387896279\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.39e6 - 4.14e6i)T \) |
| 7 | \( 1 + (1.40e12 - 1.11e12i)T \) |
| good | 2 | \( 1 + (2.96e3 + 5.14e3i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-3.53e9 - 6.12e9i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-1.86e14 + 3.22e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 9.69e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (3.20e17 - 5.55e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.96e18 + 5.13e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-3.56e19 - 6.17e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 5.37e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-1.26e21 + 2.18e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-2.70e22 - 4.67e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 1.94e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 5.40e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (1.62e24 + 2.80e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (5.34e24 - 9.25e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (3.72e25 - 6.44e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-5.19e25 - 8.98e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (-1.19e26 + 2.06e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 6.63e25T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-5.26e26 + 9.11e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (7.58e25 + 1.31e26i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 8.44e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-5.04e27 - 8.74e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 8.95e27T + 4.13e57T^{2} \) |
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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
−11.85263143710223276873411737413, −10.82080416599176395444510603717, −9.881906177499615503390422845862, −8.924344986373906807387678880010, −6.77180583591691792238623200718, −6.05860809360801523725031732052, −4.80649137841516109323518696056, −3.06070098545299494193449555641, −2.16573506557807003231239025468, −0.58077800103900323165661360547,
0.52351921015190970896047959520, 1.98000743415973577629789201940, 3.19309205551184293713109305242, 4.62788809632405521580606226782, 6.30715467385841816383532202330, 7.07497763897973837514177036943, 8.233471970282770351348014539188, 9.597971069233588922839939202957, 11.04178004468960059098529432699, 12.49193446647876720461763169285
