L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−1.12 − 0.541i)5-s + (−0.900 + 0.433i)6-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−1.12 + 0.541i)10-s + (−0.277 + 0.347i)11-s + (−0.222 + 0.974i)12-s + (−0.222 + 0.974i)14-s + (0.777 + 0.974i)15-s + (−0.900 + 0.433i)16-s + 18-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−1.12 − 0.541i)5-s + (−0.900 + 0.433i)6-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−1.12 + 0.541i)10-s + (−0.277 + 0.347i)11-s + (−0.222 + 0.974i)12-s + (−0.222 + 0.974i)14-s + (0.777 + 0.974i)15-s + (−0.900 + 0.433i)16-s + 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08736413515\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08736413515\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.623 + 0.781i)T \) |
| 3 | \( 1 + (0.900 + 0.433i)T \) |
| 7 | \( 1 + (0.900 - 0.433i)T \) |
good | 5 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + 1.80T + T^{2} \) |
| 37 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.445 + 1.94i)T + (-0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (1.80 - 0.867i)T + (0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−9.544892738448915738080920358262, −8.709346471705955633629035500129, −7.53140914998318925368785271278, −6.74378767007787960120172109797, −5.74799558860765209417712241318, −5.04261713519169634026814022431, −4.13442804599470177122820529409, −3.17267122554581779714080885191, −1.73475326122063321593680005530, −0.06851505592944514004000200950,
3.08843554310622611091191879486, 3.81683934920932558574789314126, 4.49185157645943920122146376898, 5.68843691522698953680523399443, 6.29485524478870770108173531260, 7.26380822269794891398164568092, 7.62549813323879311341836212244, 8.891092282298045647007144141602, 9.759227319809836276713423549841, 10.81330404060048118484318462213