| L(s) = 1 | − 2·3-s + 4·7-s + 2·9-s + 2·11-s − 4·17-s + 2·19-s − 8·21-s + 2·23-s − 6·27-s − 8·29-s − 8·31-s − 4·33-s − 18·37-s + 6·41-s − 14·43-s − 10·47-s + 3·49-s + 8·51-s + 2·53-s − 4·57-s + 2·59-s − 10·61-s + 8·63-s − 8·67-s − 4·69-s + 20·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 2/3·9-s + 0.603·11-s − 0.970·17-s + 0.458·19-s − 1.74·21-s + 0.417·23-s − 1.15·27-s − 1.48·29-s − 1.43·31-s − 0.696·33-s − 2.95·37-s + 0.937·41-s − 2.13·43-s − 1.45·47-s + 3/7·49-s + 1.12·51-s + 0.274·53-s − 0.529·57-s + 0.260·59-s − 1.28·61-s + 1.00·63-s − 0.977·67-s − 0.481·69-s + 2.37·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 18 T + 150 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 151 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 49 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.904368050883700745457731675852, −7.892148717967435831448672672155, −7.41909736175984779283204655543, −6.88325954915712794237007694611, −6.61057063340420348690222517315, −6.58860811594197733858941890979, −5.62861298481214014905544893733, −5.62226716006349230220381804706, −5.13876635335910296418718751260, −4.98573889412832403100517820371, −4.59704453577222032522143813760, −3.92652034911943190596946497747, −3.69583351099918363372681526471, −3.36274414035517020699179207902, −2.48676680619632877826710033961, −1.90300102550975144083961463633, −1.57507280857170281320425661679, −1.32810705984014529408994931640, 0, 0,
1.32810705984014529408994931640, 1.57507280857170281320425661679, 1.90300102550975144083961463633, 2.48676680619632877826710033961, 3.36274414035517020699179207902, 3.69583351099918363372681526471, 3.92652034911943190596946497747, 4.59704453577222032522143813760, 4.98573889412832403100517820371, 5.13876635335910296418718751260, 5.62226716006349230220381804706, 5.62861298481214014905544893733, 6.58860811594197733858941890979, 6.61057063340420348690222517315, 6.88325954915712794237007694611, 7.41909736175984779283204655543, 7.892148717967435831448672672155, 7.904368050883700745457731675852
