| L(s) = 1 | + 2·2-s − 4-s − 5-s + 2·7-s − 8·8-s − 2·10-s + 2·13-s + 4·14-s − 7·16-s − 9·17-s + 3·19-s + 20-s − 5·23-s + 2·25-s + 4·26-s − 2·28-s + 4·29-s + 31-s + 14·32-s − 18·34-s − 2·35-s − 13·37-s + 6·38-s + 8·40-s + 15·41-s + 2·43-s − 10·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 0.447·5-s + 0.755·7-s − 2.82·8-s − 0.632·10-s + 0.554·13-s + 1.06·14-s − 7/4·16-s − 2.18·17-s + 0.688·19-s + 0.223·20-s − 1.04·23-s + 2/5·25-s + 0.784·26-s − 0.377·28-s + 0.742·29-s + 0.179·31-s + 2.47·32-s − 3.08·34-s − 0.338·35-s − 2.13·37-s + 0.973·38-s + 1.26·40-s + 2.34·41-s + 0.304·43-s − 1.47·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13 T + 85 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 127 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 150 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 167 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.82154824534806689660561679242, −7.32423241876711673310991815241, −6.70794449335318096611845171806, −6.66884107732389934743177172683, −6.10604386418665054986845494698, −5.95156835846325972273579460381, −5.33184430729071749587705649504, −5.23308400772967162479985355026, −4.80962391416652266702675568173, −4.38719186111256300253134027500, −4.11511086468898737797249410513, −4.10715792636260147925833717577, −3.45797598102189069252921075057, −3.12216389846517109967095413829, −2.58422150706532836334733432265, −2.33401736636071433990273582372, −1.50005014546387321618044171023, −1.04833811416950950646002355805, 0, 0,
1.04833811416950950646002355805, 1.50005014546387321618044171023, 2.33401736636071433990273582372, 2.58422150706532836334733432265, 3.12216389846517109967095413829, 3.45797598102189069252921075057, 4.10715792636260147925833717577, 4.11511086468898737797249410513, 4.38719186111256300253134027500, 4.80962391416652266702675568173, 5.23308400772967162479985355026, 5.33184430729071749587705649504, 5.95156835846325972273579460381, 6.10604386418665054986845494698, 6.66884107732389934743177172683, 6.70794449335318096611845171806, 7.32423241876711673310991815241, 7.82154824534806689660561679242
