| L(s) = 1 | + (−3 + 4i)5-s − 9·9-s − 24i·13-s − 16i·17-s + (−7 − 24i)25-s − 42·29-s − 24i·37-s − 18·41-s + (27 − 36i)45-s − 49·49-s + 56i·53-s − 22·61-s + (96 + 72i)65-s − 96i·73-s + 81·81-s + ⋯ |
| L(s) = 1 | + (−0.600 + 0.800i)5-s − 9-s − 1.84i·13-s − 0.941i·17-s + (−0.280 − 0.959i)25-s − 1.44·29-s − 0.648i·37-s − 0.439·41-s + (0.599 − 0.800i)45-s − 0.999·49-s + 1.05i·53-s − 0.360·61-s + (1.47 + 1.10i)65-s − 1.31i·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.223259 - 0.446519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223259 - 0.446519i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (3 - 4i)T \) |
| good | 3 | \( 1 + 9T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 24iT - 169T^{2} \) |
| 17 | \( 1 + 16iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 + 42T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 24iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 56iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 96iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 78T + 7.92e3T^{2} \) |
| 97 | \( 1 - 144iT - 9.40e3T^{2} \) |
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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
−11.07679843073169300955439122055, −10.37675594230002680934789415616, −9.193351986415369135320928458234, −8.038236998522122541384528531766, −7.42300653547576834441962528391, −6.11876707951641688659589284227, −5.17083537092369082762678128908, −3.53001999551628349238889505434, −2.69836539084642045863761966770, −0.22780806880820372631024494229,
1.78143925392723425471090156169, 3.60038582962578044505787736345, 4.61696418290237672315656253195, 5.79059323876613339997269819176, 6.93265509661542304115278975975, 8.161246951428269023864119741800, 8.832809600214099944413828536001, 9.676197676310257093608065722277, 11.18083667568508571634785063036, 11.60562292940588339280598805398
