| L(s) = 1 | − 2·2-s + 3·4-s − 3·5-s − 4·7-s − 4·8-s + 6·10-s − 3·11-s − 5·13-s + 8·14-s + 5·16-s + 3·17-s − 5·19-s − 9·20-s + 6·22-s − 3·23-s + 5·25-s + 10·26-s − 12·28-s − 3·29-s − 8·31-s − 6·32-s − 6·34-s + 12·35-s + 7·37-s + 10·38-s + 12·40-s − 9·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.34·5-s − 1.51·7-s − 1.41·8-s + 1.89·10-s − 0.904·11-s − 1.38·13-s + 2.13·14-s + 5/4·16-s + 0.727·17-s − 1.14·19-s − 2.01·20-s + 1.27·22-s − 0.625·23-s + 25-s + 1.96·26-s − 2.26·28-s − 0.557·29-s − 1.43·31-s − 1.06·32-s − 1.02·34-s + 2.02·35-s + 1.15·37-s + 1.62·38-s + 1.89·40-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−10.88709375465246007266692109328, −10.60375925062863853851680724064, −10.12767765604517983346958768164, −9.836127976125770262399389548979, −9.193245987751058934679503617583, −9.045628396700969285927040373530, −8.093802168587464372059090740728, −8.047058264937251934934417284852, −7.32697019660382802746618633381, −7.31790371965350425379938717046, −6.41352881130203080659637104353, −6.21773708928578797316902363844, −5.31324951697940326793430517168, −4.71425038901120794203073840691, −3.80959599410311598621617583258, −3.28084061081759045993439417565, −2.74102370887630031957619182797, −1.85724875136122169554801817291, 0, 0,
1.85724875136122169554801817291, 2.74102370887630031957619182797, 3.28084061081759045993439417565, 3.80959599410311598621617583258, 4.71425038901120794203073840691, 5.31324951697940326793430517168, 6.21773708928578797316902363844, 6.41352881130203080659637104353, 7.31790371965350425379938717046, 7.32697019660382802746618633381, 8.047058264937251934934417284852, 8.093802168587464372059090740728, 9.045628396700969285927040373530, 9.193245987751058934679503617583, 9.836127976125770262399389548979, 10.12767765604517983346958768164, 10.60375925062863853851680724064, 10.88709375465246007266692109328
