| L(s) = 1 | − 6.92i·5-s + 12i·7-s + 6.92·11-s − 13.8i·13-s − 14·17-s − 34.6·19-s + 24i·23-s − 22.9·25-s − 34.6i·29-s − 12i·31-s + 83.1·35-s + 27.7i·37-s + 14·41-s − 6.92·43-s − 72i·47-s + ⋯ |
| L(s) = 1 | − 1.38i·5-s + 1.71i·7-s + 0.629·11-s − 1.06i·13-s − 0.823·17-s − 1.82·19-s + 1.04i·23-s − 0.919·25-s − 1.19i·29-s − 0.387i·31-s + 2.37·35-s + 0.748i·37-s + 0.341·41-s − 0.161·43-s − 1.53i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2543467715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2543467715\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6.92iT - 25T^{2} \) |
| 7 | \( 1 - 12iT - 49T^{2} \) |
| 11 | \( 1 - 6.92T + 121T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 + 14T + 289T^{2} \) |
| 19 | \( 1 + 34.6T + 361T^{2} \) |
| 23 | \( 1 - 24iT - 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 + 12iT - 961T^{2} \) |
| 37 | \( 1 - 27.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 14T + 1.68e3T^{2} \) |
| 43 | \( 1 + 6.92T + 1.84e3T^{2} \) |
| 47 | \( 1 + 72iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 62.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 48.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 55.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 90.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 50T + 5.32e3T^{2} \) |
| 79 | \( 1 + 12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 20.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 62T + 7.92e3T^{2} \) |
| 97 | \( 1 + 146T + 9.40e3T^{2} \) |
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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
−9.003267415061328035384863068489, −8.576000385933359522699652092404, −7.892274891438082513419461921832, −6.38071637180955366143481168663, −5.76057407647167316015459949882, −4.95899227945489607510700670951, −4.07797998258258538752442754497, −2.62834116320571916426900803092, −1.63473733850677570280529901935, −0.07210398363851770413562992426,
1.63722128788802585918731814736, 2.87087492515470037426214589814, 4.09411953096704024660254226080, 4.38710679026083176891820221399, 6.25771656558400054110584313818, 6.84842176635095878118898460225, 7.12672918547377474207427328763, 8.351791880302003244048213455449, 9.260962630917516419823028689461, 10.28447901670425573620246554510
