| L(s) = 1 | + (3.93e3 + 6.82e3i)2-s + (−2.39e6 + 4.14e6i)3-s + (2.37e8 − 4.11e8i)4-s + (1.52e8 + 2.64e8i)5-s − 3.76e10·6-s + (1.72e12 + 5.01e11i)7-s + (7.96e12 + 0.000244i)8-s + (−1.14e13 − 1.98e13i)9-s + (−1.20e12 + 2.08e12i)10-s + (9.83e13 − 1.70e14i)11-s + (1.13e15 + 1.96e15i)12-s − 1.06e16·13-s + (3.36e15 + 1.37e16i)14-s − 1.46e15·15-s + (−9.60e16 − 1.66e17i)16-s + (−5.25e17 + 9.09e17i)17-s + ⋯ |
| L(s) = 1 | + (0.169 + 0.294i)2-s + (−0.288 + 0.499i)3-s + (0.442 − 0.765i)4-s + (0.0112 + 0.0194i)5-s − 0.196·6-s + (0.960 + 0.279i)7-s + 0.640·8-s + (−0.166 − 0.288i)9-s + (−0.00380 + 0.00659i)10-s + (0.0780 − 0.135i)11-s + (0.255 + 0.442i)12-s − 0.748·13-s + (0.0808 + 0.330i)14-s − 0.0129·15-s + (−0.333 − 0.577i)16-s + (−0.756 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.672562627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672562627\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-1.72e12 - 5.01e11i)T \) |
| good | 2 | \( 1 + (-3.93e3 - 6.82e3i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-1.52e8 - 2.64e8i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-9.83e13 + 1.70e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 1.06e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (5.25e17 - 9.09e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-1.20e18 - 2.08e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (7.87e18 + 1.36e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 + 2.50e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (3.67e21 - 6.37e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (3.97e21 + 6.88e21i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 1.73e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 4.31e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-1.17e24 - 2.02e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-2.60e24 + 4.52e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (2.68e25 - 4.65e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (9.55e24 + 1.65e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (8.80e25 - 1.52e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 9.89e25T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-3.63e26 + 6.30e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-8.41e26 - 1.45e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 1.14e28T + 4.50e55T^{2} \) |
| 89 | \( 1 + (1.10e27 + 1.90e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 7.92e28T + 4.13e57T^{2} \) |
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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
−12.44078141075485286152455510496, −11.11695998315121631617300843927, −10.41554822048612929105169767177, −8.979061701201499281437470496773, −7.52054274402998329336187755403, −6.12808007451599522552218386137, −5.21443203790460424139842372272, −4.15608806742278825892386676497, −2.29206653491612724200442902427, −1.25436156686588498163899316473,
0.32147888934919252832372352399, 1.73992862939733450473358483112, 2.62102554019403121988972071138, 4.15409814769467473002840890196, 5.31281843586010662488182452915, 7.15777930536489398606690920959, 7.59094070762845226840560012900, 9.192674361867057364261353132150, 11.07886342964457555879538761026, 11.57276650860003227035179055861
