| L(s) = 1 | − 2·3-s − 2·4-s − 2·5-s − 2·7-s + 3·9-s + 2·11-s + 4·12-s + 4·13-s + 4·15-s + 2·17-s − 2·19-s + 4·20-s + 4·21-s − 2·23-s + 3·25-s − 4·27-s + 4·28-s + 6·29-s + 4·31-s − 4·33-s + 4·35-s − 6·36-s − 4·37-s − 8·39-s + 8·41-s − 10·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 0.894·5-s − 0.755·7-s + 9-s + 0.603·11-s + 1.15·12-s + 1.10·13-s + 1.03·15-s + 0.485·17-s − 0.458·19-s + 0.894·20-s + 0.872·21-s − 0.417·23-s + 3/5·25-s − 0.769·27-s + 0.755·28-s + 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.676·35-s − 36-s − 0.657·37-s − 1.28·39-s + 1.24·41-s − 1.52·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8268723484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8268723484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 26 T + 285 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 115 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 156 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 14 T + 177 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 185 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.900582590763942512380926127677, −9.852461270873464615123075745317, −8.966388619124155708189401254919, −8.897186008981605879157230099041, −8.368191384666051087933540030508, −8.160467642174212165676544948325, −7.37279330383101530119500603609, −6.98963571312339861262062748431, −6.68764975891301449682139812426, −6.21706954108423051533508463304, −5.66873358087859987248356378224, −5.49734949891421853928581309596, −4.59286799291106978557856143906, −4.50860217707260220428866815730, −3.78998316776499646749278114644, −3.75935684225144076739711708886, −2.95138526931300296985986346485, −2.06472346318772810908291847475, −0.926626054464907288495888226959, −0.60723918796020736205255592239,
0.60723918796020736205255592239, 0.926626054464907288495888226959, 2.06472346318772810908291847475, 2.95138526931300296985986346485, 3.75935684225144076739711708886, 3.78998316776499646749278114644, 4.50860217707260220428866815730, 4.59286799291106978557856143906, 5.49734949891421853928581309596, 5.66873358087859987248356378224, 6.21706954108423051533508463304, 6.68764975891301449682139812426, 6.98963571312339861262062748431, 7.37279330383101530119500603609, 8.160467642174212165676544948325, 8.368191384666051087933540030508, 8.897186008981605879157230099041, 8.966388619124155708189401254919, 9.852461270873464615123075745317, 9.900582590763942512380926127677
