L(s) = 1 | + (−4.57 + 2.63i)2-s + (4.83 + 1.91i)3-s + (9.93 − 17.2i)4-s + (−2.5 − 4.33i)5-s + (−27.1 + 3.99i)6-s + (−17.5 − 5.84i)7-s + 62.6i·8-s + (19.6 + 18.4i)9-s + (22.8 + 13.1i)10-s + (19.7 + 11.4i)11-s + (80.9 − 64.1i)12-s + 81.4i·13-s + (95.7 − 19.6i)14-s + (−3.78 − 25.7i)15-s + (−85.9 − 148. i)16-s + (−61.9 + 107. i)17-s + ⋯ |
L(s) = 1 | + (−1.61 + 0.933i)2-s + (0.929 + 0.368i)3-s + (1.24 − 2.15i)4-s + (−0.223 − 0.387i)5-s + (−1.84 + 0.272i)6-s + (−0.948 − 0.315i)7-s + 2.77i·8-s + (0.728 + 0.685i)9-s + (0.722 + 0.417i)10-s + (0.542 + 0.313i)11-s + (1.94 − 1.54i)12-s + 1.73i·13-s + (1.82 − 0.375i)14-s + (−0.0651 − 0.442i)15-s + (−1.34 − 2.32i)16-s + (−0.884 + 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.193245 + 0.657453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.193245 + 0.657453i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.83 - 1.91i)T \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
| 7 | \( 1 + (17.5 + 5.84i)T \) |
good | 2 | \( 1 + (4.57 - 2.63i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-19.7 - 11.4i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 81.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (61.9 - 107. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (40.3 - 23.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-98.1 + 56.6i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 80.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (15.7 + 9.06i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (49.5 + 85.7i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-60.6 - 105. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (200. + 115. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (3.91 - 6.77i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-158. + 91.6i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (141. - 244. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 521. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-77.7 - 44.8i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (298. + 517. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 249.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-147. - 256. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 711. iT - 9.12e5T^{2} \) |
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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
−14.20695022495807425616904333556, −12.79818346649065106231838430054, −11.00046033731787343899387677983, −9.993722001286644880528874868706, −9.040516892026406477815131149396, −8.629034337924555052467554854831, −7.19284280140987893944392887397, −6.42850669237812838602959620701, −4.21597189435462349857553714338, −1.76986841786520581488918630171,
0.57332089592461766302916744247, 2.60163641047063043982144673085, 3.34464888634603903369226454605, 6.72324547952454474727624760338, 7.65386052451512674367138176370, 8.776420835123757949928029287199, 9.460811960458410350171618733243, 10.45619678401992677355165968192, 11.59821920027834436818252771252, 12.65109025017533520884210686513