L(s) = 1 | − 104. i·3-s − 4.60e3·5-s + 2.39e4i·7-s + 4.81e4·9-s + 2.16e5i·11-s + 6.41e5·13-s + 4.81e5i·15-s + 9.24e5·17-s + 2.63e6i·19-s + 2.50e6·21-s + 2.37e6i·23-s + 1.14e7·25-s − 1.12e7i·27-s + 6.16e6·29-s + 2.62e7i·31-s + ⋯ |
L(s) = 1 | − 0.430i·3-s − 1.47·5-s + 1.42i·7-s + 0.814·9-s + 1.34i·11-s + 1.72·13-s + 0.633i·15-s + 0.650·17-s + 1.06i·19-s + 0.612·21-s + 0.368i·23-s + 1.17·25-s − 0.780i·27-s + 0.300·29-s + 0.917i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(2.131032443\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.131032443\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 104. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 4.60e3T + 9.76e6T^{2} \) |
| 7 | \( 1 - 2.39e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.16e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 6.41e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 9.24e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 2.63e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 2.37e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 6.16e6T + 4.20e14T^{2} \) |
| 31 | \( 1 - 2.62e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 2.49e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 2.23e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.18e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 1.78e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 1.65e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 2.04e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.00e9T + 7.13e17T^{2} \) |
| 67 | \( 1 + 1.16e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.04e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 5.20e8T + 4.29e18T^{2} \) |
| 79 | \( 1 - 4.06e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 4.48e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 4.01e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 7.96e9T + 7.37e19T^{2} \) |
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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
−10.59606672486452253223740625937, −9.393877599598554383766017211310, −8.348364760896850092989392401996, −7.70358953397964316840709335592, −6.70277016092606347862941486484, −5.56270373260236767059671210250, −4.27279219559697279559745067865, −3.45178716072356659410774285761, −1.99968444837816834011417897867, −1.02315439598914804058279541078,
0.61353691197195970411995015027, 0.974736564868131528016433149168, 3.28730410946380494080511671081, 3.87212065576085890563643482507, 4.54174617018111555552154858017, 6.16519101036005014061407443183, 7.26670801452748949038276587719, 8.000187963085628683889350926444, 8.908070216706264486869518092266, 10.26394366134956608975246929908