L(s) = 1 | + (−1.99 − 0.178i)2-s − 5.14i·3-s + (3.93 + 0.709i)4-s + (−3.67 − 3.39i)5-s + (−0.915 + 10.2i)6-s + (3.38 + 6.12i)7-s + (−7.71 − 2.11i)8-s − 17.4·9-s + (6.71 + 7.41i)10-s + 18.5i·11-s + (3.64 − 20.2i)12-s + 12.0i·13-s + (−5.65 − 12.8i)14-s + (−17.4 + 18.8i)15-s + (14.9 + 5.58i)16-s − 19.5·17-s + ⋯ |
L(s) = 1 | + (−0.996 − 0.0890i)2-s − 1.71i·3-s + (0.984 + 0.177i)4-s + (−0.734 − 0.678i)5-s + (−0.152 + 1.70i)6-s + (0.483 + 0.875i)7-s + (−0.964 − 0.264i)8-s − 1.93·9-s + (0.671 + 0.741i)10-s + 1.68i·11-s + (0.304 − 1.68i)12-s + 0.925i·13-s + (−0.404 − 0.914i)14-s + (−1.16 + 1.25i)15-s + (0.937 + 0.349i)16-s − 1.15·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.269520 + 0.158369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.269520 + 0.158369i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.99 + 0.178i)T \) |
| 5 | \( 1 + (3.67 + 3.39i)T \) |
| 7 | \( 1 + (-3.38 - 6.12i)T \) |
good | 3 | \( 1 + 5.14iT - 9T^{2} \) |
| 11 | \( 1 - 18.5iT - 121T^{2} \) |
| 13 | \( 1 - 12.0iT - 169T^{2} \) |
| 17 | \( 1 + 19.5T + 289T^{2} \) |
| 19 | \( 1 + 6.87T + 361T^{2} \) |
| 23 | \( 1 + 20.3iT - 529T^{2} \) |
| 29 | \( 1 - 2.31iT - 841T^{2} \) |
| 31 | \( 1 + 6.49iT - 961T^{2} \) |
| 37 | \( 1 - 30.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 48.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 70.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 68.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 14.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 23.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 65.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 38.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 66.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 49.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 30.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 36.8T + 9.40e3T^{2} \) |
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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
−11.77238256608189651891675487228, −11.31191014679078188018769195484, −9.574276364188088922729730949861, −8.600291959798711744161948137100, −8.055124665114984329681513971174, −7.05612654280594306386954867467, −6.41858809751537995138155431988, −4.70301009376629618360466265480, −2.34480387612418787088107187659, −1.58072641177013094171926554437,
0.21021154465022058441143635115, 3.06953161824988348723375271061, 3.89181096320284471338117735344, 5.35976082220815871481258615653, 6.64037340948876560215590699402, 8.029554383258335613234457493476, 8.565150153574051151671424328372, 9.750736950519819663421743747294, 10.66994335403272060957947797736, 10.99577942493859836471731183728