L(s) = 1 | + (−1.44 + 1.38i)2-s + (0.936 + 2.26i)3-s + (0.175 − 3.99i)4-s + (−3.18 + 7.68i)5-s + (−4.47 − 1.97i)6-s + (3.67 − 3.67i)7-s + (5.27 + 6.01i)8-s + (2.12 − 2.12i)9-s + (−6.02 − 15.5i)10-s + (6.10 − 14.7i)11-s + (9.19 − 3.34i)12-s + (2.82 + 6.80i)13-s + (−0.228 + 10.3i)14-s − 20.3·15-s + (−15.9 − 1.40i)16-s + 3.67i·17-s + ⋯ |
L(s) = 1 | + (−0.722 + 0.691i)2-s + (0.312 + 0.753i)3-s + (0.0438 − 0.999i)4-s + (−0.636 + 1.53i)5-s + (−0.746 − 0.328i)6-s + (0.524 − 0.524i)7-s + (0.659 + 0.752i)8-s + (0.236 − 0.236i)9-s + (−0.602 − 1.55i)10-s + (0.554 − 1.33i)11-s + (0.766 − 0.278i)12-s + (0.216 + 0.523i)13-s + (−0.0162 + 0.742i)14-s − 1.35·15-s + (−0.996 − 0.0876i)16-s + 0.215i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0865 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0865 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.524934 + 0.572497i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.524934 + 0.572497i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.44 - 1.38i)T \) |
good | 3 | \( 1 + (-0.936 - 2.26i)T + (-6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (3.18 - 7.68i)T + (-17.6 - 17.6i)T^{2} \) |
| 7 | \( 1 + (-3.67 + 3.67i)T - 49iT^{2} \) |
| 11 | \( 1 + (-6.10 + 14.7i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (-2.82 - 6.80i)T + (-119. + 119. i)T^{2} \) |
| 17 | \( 1 - 3.67iT - 289T^{2} \) |
| 19 | \( 1 + (1.65 - 0.686i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (8.31 + 8.31i)T + 529iT^{2} \) |
| 29 | \( 1 + (38.8 - 16.0i)T + (594. - 594. i)T^{2} \) |
| 31 | \( 1 + 4.11iT - 961T^{2} \) |
| 37 | \( 1 + (-19.8 + 47.9i)T + (-968. - 968. i)T^{2} \) |
| 41 | \( 1 + (-21.1 + 21.1i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (0.102 - 0.247i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 39.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-22.6 - 9.36i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (101. + 41.9i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (14.0 - 5.81i)T + (2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (-3.67 - 8.87i)T + (-3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (-75.7 + 75.7i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (29.0 - 29.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 2.76T + 6.24e3T^{2} \) |
| 83 | \( 1 + (79.1 - 32.8i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-72.4 - 72.4i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 - 66.0T + 9.40e3T^{2} \) |
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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
−16.71162638351974764329746013438, −15.71671200812563815547532030608, −14.64430706216752771302602774906, −14.14488219327480304980575596019, −11.22260457668720852179907644838, −10.63184348987180085492561064882, −9.144036888668808167156791654430, −7.66388780678388804145311632497, −6.38614068931851169833193961490, −3.84693200445681951709157948748,
1.62170832148395421389773165226, 4.54500816213483588616782769396, 7.54196096298141498350594402178, 8.403124929827612822797225347113, 9.625233916614435035789254969224, 11.62875921361047073809445527418, 12.47227709710950592610928955119, 13.23883724636955437420880399448, 15.32782022842986957666888222666, 16.58893933085540768437174364415