| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 4·13-s + 5·16-s − 4·17-s − 8·19-s − 4·23-s + 8·26-s + 4·29-s − 8·31-s − 6·32-s + 8·34-s + 4·37-s + 16·38-s + 12·41-s + 8·43-s + 8·46-s − 8·47-s − 12·52-s − 12·53-s − 8·58-s + 8·59-s − 12·61-s + 16·62-s + 7·64-s − 16·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1.10·13-s + 5/4·16-s − 0.970·17-s − 1.83·19-s − 0.834·23-s + 1.56·26-s + 0.742·29-s − 1.43·31-s − 1.06·32-s + 1.37·34-s + 0.657·37-s + 2.59·38-s + 1.87·41-s + 1.21·43-s + 1.17·46-s − 1.16·47-s − 1.66·52-s − 1.64·53-s − 1.05·58-s + 1.04·59-s − 1.53·61-s + 2.03·62-s + 7/8·64-s − 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.717718373277404190025356577704, −8.533734760737863034578799707392, −7.87618994658285986383645074624, −7.73070092839704685654170864529, −7.44332251937665533548020429985, −6.86112423682565169131678281474, −6.57329622933745417378931154565, −6.11629829699109230021056174201, −5.91739542907667777964753031119, −5.33105347712169935915063937345, −4.51157008383186437658852498779, −4.49658505160864412299871881077, −3.95717695750330140891634382471, −3.18847522176192251148683870816, −2.63751221012855661109858627369, −2.31063928177582113126201221757, −1.84827180968750081831556554096, −1.20497690144739431747906417288, 0, 0,
1.20497690144739431747906417288, 1.84827180968750081831556554096, 2.31063928177582113126201221757, 2.63751221012855661109858627369, 3.18847522176192251148683870816, 3.95717695750330140891634382471, 4.49658505160864412299871881077, 4.51157008383186437658852498779, 5.33105347712169935915063937345, 5.91739542907667777964753031119, 6.11629829699109230021056174201, 6.57329622933745417378931154565, 6.86112423682565169131678281474, 7.44332251937665533548020429985, 7.73070092839704685654170864529, 7.87618994658285986383645074624, 8.533734760737863034578799707392, 8.717718373277404190025356577704
