| L(s) = 1 | + (0.948 − 1.04i)2-s + (−0.796 + 0.199i)3-s + (−0.201 − 1.98i)4-s + (−2.23 − 1.05i)5-s + (−0.545 + 1.02i)6-s + (−0.443 − 1.46i)7-s + (−2.27 − 1.67i)8-s + (−2.05 + 1.09i)9-s + (−3.23 + 1.34i)10-s + (0.660 − 0.0979i)11-s + (0.557 + 1.54i)12-s + (1.50 − 4.21i)13-s + (−1.95 − 0.921i)14-s + (1.99 + 0.396i)15-s + (−3.91 + 0.803i)16-s + (0.516 − 0.102i)17-s + ⋯ |
| L(s) = 1 | + (0.670 − 0.741i)2-s + (−0.459 + 0.115i)3-s + (−0.100 − 0.994i)4-s + (−1.00 − 0.473i)5-s + (−0.222 + 0.418i)6-s + (−0.167 − 0.553i)7-s + (−0.805 − 0.592i)8-s + (−0.683 + 0.365i)9-s + (−1.02 + 0.425i)10-s + (0.199 − 0.0295i)11-s + (0.160 + 0.445i)12-s + (0.418 − 1.16i)13-s + (−0.522 − 0.246i)14-s + (0.514 + 0.102i)15-s + (−0.979 + 0.200i)16-s + (0.125 − 0.0249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.210213 - 0.964534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.210213 - 0.964534i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.948 + 1.04i)T \) |
| good | 3 | \( 1 + (0.796 - 0.199i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (2.23 + 1.05i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.443 + 1.46i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-0.660 + 0.0979i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-1.50 + 4.21i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-0.516 + 0.102i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-0.468 - 0.424i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-4.23 + 0.417i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (4.00 + 5.39i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-8.89 - 3.68i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.280 + 5.71i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (5.74 - 7.00i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-0.718 + 2.86i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-8.21 - 5.48i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (1.78 - 2.40i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (3.14 + 8.80i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (1.95 - 3.26i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-2.51 - 1.50i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-1.19 + 2.23i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-0.217 + 0.718i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (8.34 + 12.4i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.259 - 5.28i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-0.0896 - 0.00883i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-3.08 + 7.45i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.63069516494056288074812671398, −10.89998287760791827614887884925, −10.11466915451340748514778289347, −8.749848819675161929094147766046, −7.71479739736178882229310074453, −6.22117096177683918212308403694, −5.17481517115158775275876658027, −4.16196772340183672989897047224, −3.02235715086878421018419164982, −0.67307650820274651981089280961,
3.00882774369383254588347079378, 4.10220947250666852572685707942, 5.41073341541241713619691728105, 6.47604408664890642709302669653, 7.17545357861399883626670067302, 8.419099724777592957201215076744, 9.186325112282088517613610360428, 11.02010382805833996454010795413, 11.77246797719993213602043294208, 12.16539579960784075332985183511
