L(s) = 1 | + (−0.450 + 1.34i)2-s + (−1.59 − 1.20i)4-s + (0.739 + 1.78i)5-s + (0.385 + 0.385i)7-s + (2.33 − 1.59i)8-s + (−2.72 + 0.188i)10-s + (2.36 + 5.70i)11-s + (−2.30 − 0.956i)13-s + (−0.690 + 0.343i)14-s + (1.08 + 3.84i)16-s − 5.05·17-s + (−1.27 + 3.07i)19-s + (0.975 − 3.74i)20-s + (−8.71 + 0.601i)22-s + (2.28 + 2.28i)23-s + ⋯ |
L(s) = 1 | + (−0.318 + 0.948i)2-s + (−0.797 − 0.603i)4-s + (0.330 + 0.798i)5-s + (0.145 + 0.145i)7-s + (0.825 − 0.564i)8-s + (−0.862 + 0.0594i)10-s + (0.712 + 1.72i)11-s + (−0.640 − 0.265i)13-s + (−0.184 + 0.0917i)14-s + (0.271 + 0.962i)16-s − 1.22·17-s + (−0.292 + 0.705i)19-s + (0.218 − 0.836i)20-s + (−1.85 + 0.128i)22-s + (0.476 + 0.476i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.419581 + 0.914631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.419581 + 0.914631i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.450 - 1.34i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.739 - 1.78i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.385 - 0.385i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.36 - 5.70i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (2.30 + 0.956i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + (1.27 - 3.07i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.28 - 2.28i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.735 + 0.304i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 3.40iT - 31T^{2} \) |
| 37 | \( 1 + (-9.56 + 3.96i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (5.27 - 5.27i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.53 + 1.05i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 6.85iT - 47T^{2} \) |
| 53 | \( 1 + (-7.45 + 3.08i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (6.14 - 2.54i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.67 + 6.46i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-10.2 - 4.26i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-6.37 + 6.37i)T - 71iT^{2} \) |
| 73 | \( 1 + (-9.03 - 9.03i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 + (-14.8 - 6.14i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (4.97 + 4.97i)T + 89iT^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−12.27175198812869355489780830264, −10.96783342168172848139852934181, −9.975681771140457520479354037417, −9.402941101000858387263662611063, −8.190795745618734520060318904404, −7.02580193346455012840512216133, −6.62437565548104265441791104649, −5.22730220965448799086725839883, −4.15585576271241329137927456480, −2.09112335297794020528390698898,
0.892978976657856062852026102640, 2.57527373460272786780631144006, 4.06830638232353144801075484393, 5.06926542528572728768473265232, 6.49328646745601819394393952791, 8.011193984418884545641237043963, 8.998065037693440641856957762447, 9.308407916441656047168541506059, 10.81321743655338627703254607069, 11.29047354391625814134221388044