| L(s) = 1 | + (7.41e3 − 1.28e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (1.58e8 + 2.74e8i)4-s + (−5.40e9 + 9.35e9i)5-s − 7.09e10·6-s + (−8.77e11 + 1.56e12i)7-s + 1.26e13·8-s + (−1.14e13 + 1.98e13i)9-s + (8.01e13 + 1.38e14i)10-s + (1.13e15 + 1.97e15i)11-s + (7.57e14 − 1.31e15i)12-s + 8.99e15·13-s + (1.35e16 + 2.28e16i)14-s + (5.16e16 − 4i)15-s + (8.85e15 − 1.53e16i)16-s + (−1.54e16 − 2.67e16i)17-s + ⋯ |
| L(s) = 1 | + (0.320 − 0.554i)2-s + (−0.288 − 0.499i)3-s + (0.295 + 0.511i)4-s + (−0.395 + 0.685i)5-s − 0.369·6-s + (−0.489 + 0.872i)7-s + 1.01·8-s + (−0.166 + 0.288i)9-s + (0.253 + 0.438i)10-s + (0.903 + 1.56i)11-s + (0.170 − 0.295i)12-s + 0.633·13-s + (0.326 + 0.550i)14-s + 0.457·15-s + (0.0307 − 0.0532i)16-s + (−0.0222 − 0.0385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0756 - 0.997i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.0756 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(2.476727457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.476727457\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (8.77e11 - 1.56e12i)T \) |
| good | 2 | \( 1 + (-7.41e3 + 1.28e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (5.40e9 - 9.35e9i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-1.13e15 - 1.97e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 8.99e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (1.54e16 + 2.67e16i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-1.14e18 + 1.98e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-2.43e19 + 4.21e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 + 1.64e19T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-3.66e21 - 6.35e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (1.23e22 - 2.14e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 - 1.07e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 6.42e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-1.01e24 + 1.75e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-2.03e24 - 3.52e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-3.44e23 - 5.96e23i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (3.16e25 - 5.47e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (-1.47e26 - 2.54e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 + 7.74e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-6.62e24 - 1.14e25i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.09e27 + 1.90e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 + 7.38e26T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-1.42e28 + 2.46e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 - 4.23e28T + 4.13e57T^{2} \) |
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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
−12.21563362970641305410395802455, −11.66201958365293179539245850038, −10.37280846042907827778815866619, −8.758185027689681700111192981678, −7.19268199726671009552635505289, −6.58732297800218647637649321666, −4.75543868447807236021594247349, −3.39783675267369434920818268946, −2.45676110235968700268774852298, −1.35856888023386508350900527590,
0.53974467928112361207668120988, 1.19545684763180018508353978567, 3.46824853519255969722006053682, 4.37127830603127570493806227913, 5.71682899147612047080005655184, 6.52423806817118442525050797056, 8.026631896064504290292156284651, 9.420615276239429157221338762303, 10.71656454898260544761516528704, 11.62088462385402204012629540821
