| L(s) = 1 | + 7·3-s − 14·5-s − 14·7-s + 12·9-s + 29·11-s − 94·13-s − 98·15-s − 37·17-s − 38·19-s − 98·21-s + 72·23-s − 51·25-s + 14·27-s − 77·29-s + 585·31-s + 203·33-s + 196·35-s − 110·37-s − 658·39-s − 199·41-s − 324·43-s − 168·45-s + 64·47-s + 147·49-s − 259·51-s + 613·53-s − 406·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 1.25·5-s − 0.755·7-s + 4/9·9-s + 0.794·11-s − 2.00·13-s − 1.68·15-s − 0.527·17-s − 0.458·19-s − 1.01·21-s + 0.652·23-s − 0.407·25-s + 0.0997·27-s − 0.493·29-s + 3.38·31-s + 1.07·33-s + 0.946·35-s − 0.488·37-s − 2.70·39-s − 0.758·41-s − 1.14·43-s − 0.556·45-s + 0.198·47-s + 3/7·49-s − 0.711·51-s + 1.58·53-s − 0.995·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 14 T + 247 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 29 T + 2323 T^{2} - 29 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 94 T + 6551 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 37 T + 8995 T^{2} + 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 72 T + 25513 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 77 T + 27871 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 585 T + 144979 T^{2} - 585 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 110 T + 57531 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 199 T + 26683 T^{2} + 199 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 324 T + 160090 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 64 T + 190873 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 613 T + 273133 T^{2} - 613 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 p T + 20887 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1150 T + 740855 T^{2} + 1150 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 649 T + 678717 T^{2} + 649 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 150 T + 428947 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 625 T + 283121 T^{2} + 625 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 988 T + 430562 T^{2} - 988 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1589 T + 1741651 T^{2} + 1589 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 302 T + 175366 T^{2} - 302 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 690 T + 1532479 T^{2} + 690 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.440964008413877245522013345785, −8.344582429459640704345538852097, −7.74010609306771739439139359820, −7.48795036079778525137502038682, −7.04376716798980380930251117238, −6.85198382199154608124246860665, −6.15700518001562718218840128235, −6.06787332113390786912415514488, −5.03275503433016018897213985822, −4.86204275445786664345214050361, −4.28279989542819299457587436348, −4.09601171032764689906368351958, −3.39983017234312367654861729526, −3.16787936157408066978213346918, −2.54275960939717627540060865004, −2.48843697241038541838432259947, −1.65650454078402592009380597124, −0.906518602653379954870180114352, 0, 0,
0.906518602653379954870180114352, 1.65650454078402592009380597124, 2.48843697241038541838432259947, 2.54275960939717627540060865004, 3.16787936157408066978213346918, 3.39983017234312367654861729526, 4.09601171032764689906368351958, 4.28279989542819299457587436348, 4.86204275445786664345214050361, 5.03275503433016018897213985822, 6.06787332113390786912415514488, 6.15700518001562718218840128235, 6.85198382199154608124246860665, 7.04376716798980380930251117238, 7.48795036079778525137502038682, 7.74010609306771739439139359820, 8.344582429459640704345538852097, 8.440964008413877245522013345785
