L(s) = 1 | + (0.526 + 1.31i)2-s + (−0.881 + 0.471i)3-s + (−1.44 + 1.38i)4-s + (−2.24 − 2.73i)5-s + (−1.08 − 0.909i)6-s + (3.89 − 2.60i)7-s + (−2.57 − 1.16i)8-s + (0.555 − 0.831i)9-s + (2.40 − 4.38i)10-s + (−0.156 + 0.0473i)11-s + (0.622 − 1.90i)12-s + (2.21 + 1.81i)13-s + (5.46 + 3.74i)14-s + (3.26 + 1.35i)15-s + (0.175 − 3.99i)16-s + (1.46 − 0.606i)17-s + ⋯ |
L(s) = 1 | + (0.372 + 0.928i)2-s + (−0.509 + 0.272i)3-s + (−0.722 + 0.691i)4-s + (−1.00 − 1.22i)5-s + (−0.442 − 0.371i)6-s + (1.47 − 0.983i)7-s + (−0.910 − 0.412i)8-s + (0.185 − 0.277i)9-s + (0.760 − 1.38i)10-s + (−0.0470 + 0.0142i)11-s + (0.179 − 0.548i)12-s + (0.614 + 0.504i)13-s + (1.46 + 0.999i)14-s + (0.843 + 0.349i)15-s + (0.0439 − 0.999i)16-s + (0.355 − 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14421 - 0.0721950i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14421 - 0.0721950i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.526 - 1.31i)T \) |
| 3 | \( 1 + (0.881 - 0.471i)T \) |
good | 5 | \( 1 + (2.24 + 2.73i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-3.89 + 2.60i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.156 - 0.0473i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-2.21 - 1.81i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-1.46 + 0.606i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.843 + 8.55i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (1.83 - 0.365i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 5.22i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-5.96 - 5.96i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.516 - 0.0508i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.115 - 0.582i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (8.00 + 4.28i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (3.84 + 9.27i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-3.13 - 10.3i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (2.05 - 1.68i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-2.62 - 4.90i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (4.68 + 8.76i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (4.85 + 7.26i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-4.96 - 3.31i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.986 - 2.38i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-13.1 + 1.29i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-2.92 - 0.581i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-2.72 - 2.72i)T + 97iT^{2} \) |
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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
−11.71448988447633528650665747315, −10.56056988508608423357579071836, −9.065756598175196529490594042677, −8.358059151899761290303453777684, −7.62837902201523633567095695480, −6.64689568117584820576685127435, −5.04649622601485424950230396655, −4.68464978216218643619070622681, −3.87969214044966707794410509804, −0.808130582106741473003499796415,
1.69429984413180956604846468361, 3.09043886090786185333964987841, 4.24035796763838735155783423928, 5.45513348389168991901416452129, 6.31553299167519643063372348016, 7.968207651185756868004217995957, 8.284626012827344179669448803447, 10.02163433807121332053170969794, 10.79026052710970944375347820069, 11.48433841333483107073630441633