L(s) = 1 | + (−1.32 − 0.505i)2-s + (1.48 + 1.33i)4-s + (−2.10 + 2.10i)5-s − 4.40·7-s + (−1.28 − 2.51i)8-s + (3.84 − 1.71i)10-s + (0.215 + 0.215i)11-s + (−2.73 + 2.73i)13-s + (5.82 + 2.22i)14-s + (0.430 + 3.97i)16-s + 2.36i·17-s + (0.758 + 0.758i)19-s + (−5.94 + 0.320i)20-s + (−0.175 − 0.393i)22-s − 1.75i·23-s + ⋯ |
L(s) = 1 | + (−0.933 − 0.357i)2-s + (0.744 + 0.667i)4-s + (−0.941 + 0.941i)5-s − 1.66·7-s + (−0.456 − 0.889i)8-s + (1.21 − 0.542i)10-s + (0.0650 + 0.0650i)11-s + (−0.758 + 0.758i)13-s + (1.55 + 0.595i)14-s + (0.107 + 0.994i)16-s + 0.573i·17-s + (0.174 + 0.174i)19-s + (−1.32 + 0.0717i)20-s + (−0.0374 − 0.0839i)22-s − 0.366i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.118743 + 0.245713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.118743 + 0.245713i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.32 + 0.505i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.10 - 2.10i)T - 5iT^{2} \) |
| 7 | \( 1 + 4.40T + 7T^{2} \) |
| 11 | \( 1 + (-0.215 - 0.215i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.73 - 2.73i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.36iT - 17T^{2} \) |
| 19 | \( 1 + (-0.758 - 0.758i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.75iT - 23T^{2} \) |
| 29 | \( 1 + (-5.54 - 5.54i)T + 29iT^{2} \) |
| 31 | \( 1 + 9.01iT - 31T^{2} \) |
| 37 | \( 1 + (-3.10 - 3.10i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + (3.54 - 3.54i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.90T + 47T^{2} \) |
| 53 | \( 1 + (-2.71 + 2.71i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.40 + 3.40i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.75 - 1.75i)T - 61iT^{2} \) |
| 67 | \( 1 + (-9.11 - 9.11i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 0.482iT - 73T^{2} \) |
| 79 | \( 1 - 6.88iT - 79T^{2} \) |
| 83 | \( 1 + (4.79 - 4.79i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−13.12849057893941737602943411612, −12.18321701537019838756820210526, −11.38473908331081647588123429769, −10.22834258910860332673271829976, −9.596764207037267506754349914697, −8.274337918454254630258024489549, −7.03632347333026010804459656181, −6.50604931281048086346477378294, −3.82445704783862574617619330052, −2.78944170829984523088386842377,
0.34622736850952760303374964206, 3.13988077070056694786208827310, 5.05739704342173902425332181510, 6.49311046457557813012650474911, 7.53931740352970715085164952511, 8.597535673032101057622715391606, 9.541224545159776518375224442126, 10.35917155156619707245517063971, 11.86622457892228948186243110786, 12.45885799641801747009289121019