L(s) = 1 | + (−0.0811 − 1.73i)3-s + (0.707 − 0.707i)7-s + (−2.98 + 0.280i)9-s − 2.33i·11-s + (3.22 + 3.22i)13-s + (4.22 + 4.22i)17-s + 7.12i·19-s + (−1.28 − 1.16i)21-s + (−5.87 + 5.87i)23-s + (0.728 + 5.14i)27-s + 10.6·29-s + 8.24·31-s + (−4.03 + 0.189i)33-s + (−3.62 + 3.62i)37-s + (5.31 − 5.84i)39-s + ⋯ |
L(s) = 1 | + (−0.0468 − 0.998i)3-s + (0.267 − 0.267i)7-s + (−0.995 + 0.0935i)9-s − 0.703i·11-s + (0.894 + 0.894i)13-s + (1.02 + 1.02i)17-s + 1.63i·19-s + (−0.279 − 0.254i)21-s + (−1.22 + 1.22i)23-s + (0.140 + 0.990i)27-s + 1.97·29-s + 1.48·31-s + (−0.702 + 0.0329i)33-s + (−0.595 + 0.595i)37-s + (0.851 − 0.935i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.773075462\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.773075462\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0811 + 1.73i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + 2.33iT - 11T^{2} \) |
| 13 | \( 1 + (-3.22 - 3.22i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.22 - 4.22i)T + 17iT^{2} \) |
| 19 | \( 1 - 7.12iT - 19T^{2} \) |
| 23 | \( 1 + (5.87 - 5.87i)T - 23iT^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 + (3.62 - 3.62i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.66iT - 41T^{2} \) |
| 43 | \( 1 + (4.41 + 4.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.64 + 1.64i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.87 + 5.87i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + (2.03 - 2.03i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.02iT - 71T^{2} \) |
| 73 | \( 1 + (-4.24 - 4.24i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.43iT - 79T^{2} \) |
| 83 | \( 1 + (5.87 - 5.87i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.32T + 89T^{2} \) |
| 97 | \( 1 + (0.397 - 0.397i)T - 97iT^{2} \) |
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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
−8.680106512575252194549166335484, −8.246091926883696748137824910056, −7.71982735798021719983129353094, −6.54601342807017437520226267832, −6.13388786185051522886459844485, −5.34563985629166551452539292133, −3.96701800664124273469170334546, −3.29489782232168458251071187743, −1.83720186459402785456656289928, −1.14992457426787526299227422760,
0.75278761888070431327741991250, 2.59143907509758270355996273155, 3.18796246007848856847145021706, 4.58691860256994796437424677195, 4.78913160924355955104805320866, 5.90624127717835427646731478959, 6.60950646790710329676348709992, 7.81359605933882091098204121396, 8.440868821142608271210328344576, 9.134179058391126720065497329918