| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 4·5-s + 4·6-s + 4·8-s + 3·9-s − 8·10-s − 2·11-s + 6·12-s − 8·13-s − 8·15-s + 5·16-s − 4·17-s + 6·18-s − 4·19-s − 12·20-s − 4·22-s − 4·23-s + 8·24-s + 2·25-s − 16·26-s + 4·27-s + 8·29-s − 16·30-s − 4·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.78·5-s + 1.63·6-s + 1.41·8-s + 9-s − 2.52·10-s − 0.603·11-s + 1.73·12-s − 2.21·13-s − 2.06·15-s + 5/4·16-s − 0.970·17-s + 1.41·18-s − 0.917·19-s − 2.68·20-s − 0.852·22-s − 0.834·23-s + 1.63·24-s + 2/5·25-s − 3.13·26-s + 0.769·27-s + 1.48·29-s − 2.92·30-s − 0.718·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 136 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−8.215981416779395181291769372963, −8.110407111554353029958013414770, −7.49882704435399884556820442935, −7.48923712658985638355101404706, −6.96979435446342505166150288670, −6.87071793373069035278160071436, −6.03862213101764865665080897013, −5.97029620436706895677716997473, −4.92910596935009419549359061505, −4.90445677016367894451661047716, −4.45573715805451005124604622301, −4.35447120178747222564512163752, −3.60036435677903182752770454037, −3.49416632644068465664898719074, −2.78602905574946369160953807154, −2.72038077894336095896560087732, −1.83914839946354303111103362770, −1.82051508499627481912207664278, 0, 0,
1.82051508499627481912207664278, 1.83914839946354303111103362770, 2.72038077894336095896560087732, 2.78602905574946369160953807154, 3.49416632644068465664898719074, 3.60036435677903182752770454037, 4.35447120178747222564512163752, 4.45573715805451005124604622301, 4.90445677016367894451661047716, 4.92910596935009419549359061505, 5.97029620436706895677716997473, 6.03862213101764865665080897013, 6.87071793373069035278160071436, 6.96979435446342505166150288670, 7.48923712658985638355101404706, 7.49882704435399884556820442935, 8.110407111554353029958013414770, 8.215981416779395181291769372963
