| L(s) = 1 | − 2.82i·5-s − 14.1·7-s − 20i·11-s − 39.5i·13-s + 34·17-s − 52i·19-s + 62.2·23-s + 117·25-s + 200. i·29-s − 110.·31-s + 40.0i·35-s − 271. i·37-s − 26·41-s + 252i·43-s − 345.·47-s + ⋯ |
| L(s) = 1 | − 0.252i·5-s − 0.763·7-s − 0.548i·11-s − 0.844i·13-s + 0.485·17-s − 0.627i·19-s + 0.564·23-s + 0.936·25-s + 1.28i·29-s − 0.639·31-s + 0.193i·35-s − 1.20i·37-s − 0.0990·41-s + 0.893i·43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4735166837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4735166837\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.82iT - 125T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 11 | \( 1 + 20iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 39.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 34T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 200. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 271. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 26T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 681. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 364iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 735. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 338T + 3.89e5T^{2} \) |
| 79 | \( 1 + 789.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 234T + 7.04e5T^{2} \) |
| 97 | \( 1 + 178T + 9.12e5T^{2} \) |
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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
−9.078678322942381248421876490011, −8.261549246368435166903089686421, −7.32437033103966745370128269318, −6.50262763593968272060045176338, −5.57700880814428976868814636181, −4.80639317263846726247507254275, −3.45421463787945058567909111710, −2.86893437219061693048737501733, −1.24064297640487049660214893673, −0.11922734409271183735413540236,
1.44398644820311621684666066920, 2.67386613528431030394105818079, 3.64761301963849003627339300308, 4.61542322328332252282337477321, 5.68787735160044874478662116797, 6.63135501773375096995467674486, 7.16934642358738800856645392961, 8.222287348324349810674334270868, 9.105293896188890676852900184548, 9.899770647292138890258206784916
