L(s) = 1 | + (1.82 − 0.826i)2-s + (2.63 − 3.00i)4-s + (3.61 − 6.26i)5-s + (5.95 − 3.44i)7-s + (2.31 − 7.65i)8-s + (1.41 − 14.4i)10-s + (−15.5 + 8.97i)11-s + (−3.83 + 6.64i)13-s + (8.01 − 11.1i)14-s + (−2.11 − 15.8i)16-s + 1.43·17-s + 13.8i·19-s + (−9.32 − 27.4i)20-s + (−20.9 + 29.1i)22-s + (14.3 + 8.29i)23-s + ⋯ |
L(s) = 1 | + (0.910 − 0.413i)2-s + (0.658 − 0.752i)4-s + (0.723 − 1.25i)5-s + (0.851 − 0.491i)7-s + (0.289 − 0.957i)8-s + (0.141 − 1.44i)10-s + (−1.41 + 0.816i)11-s + (−0.295 + 0.511i)13-s + (0.572 − 0.799i)14-s + (−0.131 − 0.991i)16-s + 0.0845·17-s + 0.731i·19-s + (−0.466 − 1.37i)20-s + (−0.950 + 1.32i)22-s + (0.624 + 0.360i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0211 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0211 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.23874 - 2.28655i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23874 - 2.28655i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.82 + 0.826i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-3.61 + 6.26i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-5.95 + 3.44i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (15.5 - 8.97i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (3.83 - 6.64i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 1.43T + 289T^{2} \) |
| 19 | \( 1 - 13.8iT - 361T^{2} \) |
| 23 | \( 1 + (-14.3 - 8.29i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (18.7 + 32.4i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-46.7 - 26.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 44.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-28.4 + 49.3i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (58.5 - 33.8i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-8.17 + 4.71i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 7.82T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-19.4 - 11.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-33.5 - 58.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-59.9 - 34.6i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 7.14iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 80.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-40.5 + 23.4i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (123. - 71.0i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 42.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + (31.4 + 54.4i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.33929282406327249049576790199, −10.22717760237739447308693413382, −9.649868573088014514434029497888, −8.253646808413345216455225544056, −7.25872882299148412251941226049, −5.77524859987531270745432038134, −4.96418147185366715965580215122, −4.32947475026475249773336268968, −2.39840201765888921482566479590, −1.26801998830538892209240012513,
2.42694053300384852348994950769, 3.04472548610937927951439817680, 4.88826636551815172479780668923, 5.64941150754795982368567735196, 6.59923678355478474163004878112, 7.66377554843521695513998770497, 8.456706833653557464246778799992, 10.05381856625299143598376667705, 11.00343587344759360940574854098, 11.43114906174000633803061390411