| L(s) = 1 | + (1.01e4 + 1.75e4i)2-s + (−2.39e6 + 4.14e6i)3-s + (6.38e7 − 1.10e8i)4-s + (−1.35e10 − 2.33e10i)5-s − 9.67e10·6-s + (−1.40e12 + 1.11e12i)7-s + (1.34e13 + 0.000488i)8-s + (−1.14e13 − 1.98e13i)9-s + (2.73e14 − 4.73e14i)10-s + (−1.06e15 + 1.84e15i)11-s + (3.05e14 + 5.28e14i)12-s + 7.05e15·13-s + (−3.37e16 − 1.34e16i)14-s + 1.29e17·15-s + (1.01e17 + 1.76e17i)16-s + (−3.77e17 + 6.54e17i)17-s + ⋯ |
| L(s) = 1 | + (0.436 + 0.756i)2-s + (−0.288 + 0.499i)3-s + (0.118 − 0.205i)4-s + (−0.989 − 1.71i)5-s − 0.504·6-s + (−0.785 + 0.619i)7-s + 1.08·8-s + (−0.166 − 0.288i)9-s + (0.864 − 1.49i)10-s + (−0.843 + 1.46i)11-s + (0.0686 + 0.118i)12-s + 0.496·13-s + (−0.810 − 0.323i)14-s + 1.14·15-s + (0.352 + 0.611i)16-s + (−0.544 + 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0807i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.294883919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294883919\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-1.01e4 - 1.75e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (1.35e10 + 2.33e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (1.06e15 - 1.84e15i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 - 7.05e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (3.77e17 - 6.54e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.13e18 + 3.69e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-1.12e19 - 1.94e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 + 6.55e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + (9.67e20 - 1.67e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (2.18e22 + 3.78e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 9.60e22T + 5.89e46T^{2} \) |
| 43 | \( 1 - 3.20e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-1.50e23 - 2.59e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-4.52e24 + 7.83e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-1.87e25 + 3.24e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (3.41e25 + 5.90e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (2.17e26 - 3.76e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 1.14e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-4.59e26 + 7.96e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.82e27 - 3.15e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 9.34e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (3.64e26 + 6.31e26i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 5.97e27T + 4.13e57T^{2} \) |
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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
−12.54447068646339170191680933003, −11.02204116835021277319684882354, −9.529328307221703346117942400019, −8.424061736621029149208662384139, −7.08359482453863846032475400800, −5.63082879515807584627530071785, −4.81500226104038256115291782197, −4.00002148943852169659855424959, −1.89051721724323648802555737395, −0.40525579773600459917761620994,
0.59952995178787616764412273654, 2.48484982912118980462530396013, 3.20059882081699120861821184810, 3.99500234767905470148642780052, 6.16178160344662361179535299829, 7.16154272285384268543242031375, 8.026466623538415385362720802110, 10.56633520196611068296405111790, 10.92932234840819627287095046369, 11.93069112512200250935868161117
