| L(s) = 1 | + (−1.23e4 + 2.14e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (−3.79e7 − 6.58e7i)4-s + (6.22e9 − 1.07e10i)5-s + (1.18e11 − 7.62e−6i)6-s + (−1.28e12 + 1.25e12i)7-s + (−1.14e13 + 0.000854i)8-s + (−1.14e13 + 1.98e13i)9-s + (1.54e14 + 2.67e14i)10-s + (1.80e14 + 3.11e14i)11-s + (−1.81e14 + 3.14e14i)12-s + 2.79e16·13-s + (−1.09e16 − 4.30e16i)14-s − 5.95e16·15-s + (1.61e17 − 2.79e17i)16-s + (−5.25e16 − 9.09e16i)17-s + ⋯ |
| L(s) = 1 | + (−0.534 + 0.925i)2-s + (−0.288 − 0.499i)3-s + (−0.0707 − 0.122i)4-s + (0.456 − 0.790i)5-s + 0.616·6-s + (−0.716 + 0.697i)7-s − 0.917·8-s + (−0.166 + 0.288i)9-s + (0.487 + 0.844i)10-s + (0.142 + 0.247i)11-s + (−0.0408 + 0.0707i)12-s + 1.96·13-s + (−0.262 − 1.03i)14-s − 0.526·15-s + (0.560 − 0.971i)16-s + (−0.0756 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.321161245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321161245\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (1.28e12 - 1.25e12i)T \) |
| good | 2 | \( 1 + (1.23e4 - 2.14e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-6.22e9 + 1.07e10i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-1.80e14 - 3.11e14i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 2.79e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (5.25e16 + 9.09e16i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (3.01e17 - 5.22e17i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (1.18e19 - 2.05e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 + 2.45e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (2.84e21 + 4.93e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-3.09e22 + 5.36e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 - 3.95e22T + 5.89e46T^{2} \) |
| 43 | \( 1 + 4.30e22T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-2.87e23 + 4.97e23i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-5.02e24 - 8.70e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (6.97e24 + 1.20e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (3.55e25 - 6.15e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.06e25 + 1.84e25i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 - 8.18e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (4.06e26 + 7.03e26i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.24e27 + 2.16e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 + 5.32e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (5.83e27 - 1.01e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 - 2.43e28T + 4.13e57T^{2} \) |
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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
−12.60619306150783359699534938744, −11.31965703334572359902511400361, −9.358727976585137497916248693233, −8.725415195872103455794400446177, −7.46090626527044952933807549002, −6.10695161930203579625890843039, −5.68617830485604557517506597001, −3.60958094196435615305187459688, −1.99540545415734500636462519079, −0.70865990635976465031239090475,
0.51818053850901377525399889666, 1.59832587226038509464778173171, 3.05220050857367845393591372587, 3.80861437868922035545408183071, 5.89529181319458215631450079749, 6.64335986740596001855960379231, 8.712091554274781921831821493850, 9.833871056911504502974567219934, 10.71905158648925255570078682962, 11.24599753628708974513536910185
