| L(s) = 1 | + (−2.17e4 − 3.76e4i)2-s + (−2.39e6 + 4.14e6i)3-s + (−6.78e8 + 1.17e9i)4-s + (−1.24e10 − 2.16e10i)5-s + 2.08e11·6-s + (−9.19e11 − 1.54e12i)7-s + (3.56e13 − 0.00390i)8-s + (−1.14e13 − 1.98e13i)9-s + (−5.43e14 + 9.41e14i)10-s + (4.20e14 − 7.29e14i)11-s + (−3.24e15 − 5.61e15i)12-s + 1.63e15·13-s + (−3.80e16 + 6.81e16i)14-s + 1.19e17·15-s + (−4.11e17 − 7.12e17i)16-s + (5.47e17 − 9.48e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.938 − 1.62i)2-s + (−0.288 + 0.499i)3-s + (−1.26 + 2.18i)4-s + (−0.915 − 1.58i)5-s + 1.08·6-s + (−0.512 − 0.858i)7-s + 2.86·8-s + (−0.166 − 0.288i)9-s + (−1.71 + 2.97i)10-s + (0.334 − 0.578i)11-s + (−0.729 − 1.26i)12-s + 0.115·13-s + (−0.914 + 1.63i)14-s + 1.05·15-s + (−1.42 − 2.47i)16-s + (0.788 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0486 + 0.998i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.0486 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.4519255359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4519255359\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (9.19e11 + 1.54e12i)T \) |
| good | 2 | \( 1 + (2.17e4 + 3.76e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (1.24e10 + 2.16e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-4.20e14 + 7.29e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 - 1.63e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-5.47e17 + 9.48e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.63e18 + 4.56e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-5.12e18 - 8.87e18i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 5.58e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + (3.43e21 - 5.95e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-1.75e22 - 3.03e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 2.64e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 2.22e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-3.90e23 - 6.76e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (6.32e23 - 1.09e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (3.14e25 - 5.44e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-6.00e25 - 1.04e26i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (-3.14e24 + 5.44e24i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 5.79e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-5.95e26 + 1.03e27i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.19e27 - 2.07e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 8.14e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (3.19e27 + 5.53e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 6.02e28T + 4.13e57T^{2} \) |
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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
−11.64970367536051240688636185376, −10.72250641237533752815530393129, −9.371773164358676976823703766167, −8.800161064218954158658107053856, −7.50511193443125210811752130597, −4.83684153020650483990805714365, −4.00820447948845462742456335481, −3.02292656796294578366336327700, −1.01298093036671388521984639904, −0.71114766198495894325903914911,
0.26265137308884288525317069272, 1.99479948128466530790963960729, 3.85988010209950214801024352565, 5.90289283260034129482435229370, 6.40911755938130128106155887460, 7.49866453696172998068284905561, 8.235632194590368166815196047607, 9.798641329554554666875389480459, 10.87313651216501238756973327594, 12.47509125571917874145732403225
