L(s) = 1 | + 2i·2-s − 4·4-s + 29i·7-s − 8i·8-s − 57·11-s − 20i·13-s − 58·14-s + 16·16-s − 72i·17-s + 106·19-s − 114i·22-s − 174i·23-s + 40·26-s − 116i·28-s + 210·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.56i·7-s − 0.353i·8-s − 1.56·11-s − 0.426i·13-s − 1.10·14-s + 0.250·16-s − 1.02i·17-s + 1.27·19-s − 1.10i·22-s − 1.57i·23-s + 0.301·26-s − 0.782i·28-s + 1.34·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.620226819\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.620226819\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 29iT - 343T^{2} \) |
| 11 | \( 1 + 57T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 72iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 106T + 6.85e3T^{2} \) |
| 23 | \( 1 + 174iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 210T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 474iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 81iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 84T + 2.05e5T^{2} \) |
| 61 | \( 1 - 56T + 2.26e5T^{2} \) |
| 67 | \( 1 + 142iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 360T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 160T + 4.93e5T^{2} \) |
| 83 | \( 1 + 735iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 954T + 7.04e5T^{2} \) |
| 97 | \( 1 - 191iT - 9.12e5T^{2} \) |
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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
−9.236573249697945980053961757728, −8.399949848473791044922229643162, −7.907735271204015252541322437778, −6.92356320768044654902762132567, −5.95021969662618112462855131914, −5.25525762468887170464745488654, −4.73930524845234838634584571955, −2.98386905272880015320848510951, −2.50199115792477931641273723951, −0.61479866302321414715485372095,
0.65990025893975089199837236689, 1.65801857273428066666728279876, 3.00328441577793788160038904869, 3.77535007477901229118472222940, 4.72152331123398583930310862577, 5.52818148011570479624502509953, 6.77064568934388662209729383080, 7.68446728093418935111293908007, 8.112005710953596558754537046565, 9.360476442195102426909377625070