| L(s) = 1 | + 4.92i·5-s − 2.92i·7-s − 14.9·11-s + 23.8i·13-s + 19.8·17-s + 30.9·19-s − 8i·23-s + 0.712·25-s + 16.9i·29-s − 38.6i·31-s + 14.4·35-s + 9.71i·37-s − 8.14·41-s − 5.35·43-s + 69.8i·47-s + ⋯ |
| L(s) = 1 | + 0.985i·5-s − 0.418i·7-s − 1.35·11-s + 1.83i·13-s + 1.16·17-s + 1.62·19-s − 0.347i·23-s + 0.0285·25-s + 0.583i·29-s − 1.24i·31-s + 0.412·35-s + 0.262i·37-s − 0.198·41-s − 0.124·43-s + 1.48i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.419520057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419520057\) |
| \(L(2)\) |
|
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 | \( 1 - 4.92iT - 25T^{2} \) |
| 7 | \( 1 + 2.92iT - 49T^{2} \) |
| 11 | \( 1 + 14.9T + 121T^{2} \) |
| 13 | \( 1 - 23.8iT - 169T^{2} \) |
| 17 | \( 1 - 19.8T + 289T^{2} \) |
| 19 | \( 1 - 30.9T + 361T^{2} \) |
| 23 | \( 1 + 8iT - 529T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 + 38.6iT - 961T^{2} \) |
| 37 | \( 1 - 9.71iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.14T + 1.68e3T^{2} \) |
| 43 | \( 1 + 5.35T + 1.84e3T^{2} \) |
| 47 | \( 1 - 69.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.928iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 108.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 14iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 16.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 43.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 25.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 62.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 50.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 145.T + 9.40e3T^{2} \) |
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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
−9.338025044339737578876124333760, −8.141810523518589500202146872958, −7.44539158425661592376999788083, −6.98434883691103231098192984693, −6.04210695202699525203981849797, −5.18854356762923000249755998877, −4.25292037536988507365301003005, −3.23561570943049891956125350967, −2.53822157486449683859468043156, −1.25807467975227561691568413115,
0.37431713698897301608640160536, 1.32950494704440607447298536063, 2.82583631836625586953611203762, 3.36540487703239381113515927384, 4.86518821635856353544940135363, 5.43242587593914861997942320464, 5.71611470275304737213454745951, 7.29753726177857608177618574374, 7.86107664898972817197612323918, 8.409844843644115683091304855102
