| L(s) = 1 | + 2-s + 2·3-s − 2·4-s + 2·6-s − 4·7-s − 3·8-s + 3·9-s − 6·11-s − 4·12-s − 2·13-s − 4·14-s + 16-s − 4·17-s + 3·18-s − 8·21-s − 6·22-s + 13·23-s − 6·24-s − 2·26-s + 4·27-s + 8·28-s − 5·29-s + 4·31-s + 2·32-s − 12·33-s − 4·34-s − 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s − 1.51·7-s − 1.06·8-s + 9-s − 1.80·11-s − 1.15·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 1.74·21-s − 1.27·22-s + 2.71·23-s − 1.22·24-s − 0.392·26-s + 0.769·27-s + 1.51·28-s − 0.928·29-s + 0.718·31-s + 0.353·32-s − 2.08·33-s − 0.685·34-s − 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_4$ | \( 1 + 21 T + 191 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 157 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 183 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−8.941544561304888954909567596488, −8.709866378813517048946307232499, −8.239439401933478671742089111929, −8.139730978028488726897058985380, −7.45412558900011014759405657885, −6.81692100854864692449781172805, −6.72668969607071447382370431822, −6.57252330089845428474752477339, −5.36004164896512178376169438276, −5.29858645072028312578267372402, −4.92915962404746552943019094001, −4.69030035358369051712084731662, −3.83358179241607855263554064043, −3.52621189195649390647901322075, −3.02564765773836666571548989093, −2.95888308481646293476430397970, −2.24143243840522682238543152963, −1.49019527194744572704569608914, 0, 0,
1.49019527194744572704569608914, 2.24143243840522682238543152963, 2.95888308481646293476430397970, 3.02564765773836666571548989093, 3.52621189195649390647901322075, 3.83358179241607855263554064043, 4.69030035358369051712084731662, 4.92915962404746552943019094001, 5.29858645072028312578267372402, 5.36004164896512178376169438276, 6.57252330089845428474752477339, 6.72668969607071447382370431822, 6.81692100854864692449781172805, 7.45412558900011014759405657885, 8.139730978028488726897058985380, 8.239439401933478671742089111929, 8.709866378813517048946307232499, 8.941544561304888954909567596488
