L(s) = 1 | − 3·9-s + 4·16-s − 16·19-s + 22·31-s + 22·37-s + 13·49-s − 10·67-s + 20·73-s + 28·97-s − 66·103-s + 38·109-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 48·171-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | − 9-s + 16-s − 3.67·19-s + 3.95·31-s + 3.61·37-s + 13/7·49-s − 1.22·67-s + 2.34·73-s + 2.84·97-s − 6.50·103-s + 3.63·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s + 3.67·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.510824552\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.510824552\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.624492040485525580235641918852, −8.429955814812030758060907150571, −8.413903034769275654190331708903, −7.76599749664053792588305504743, −7.75002290152873825471492779088, −7.67711288871405901084678875089, −7.05596577721133797356519805652, −6.62749018033338325467222732199, −6.51574196723334306186197570090, −6.19016589326761072017604130012, −6.13298127990956816676196586426, −5.87874239834461993758233121628, −5.64198617118707104973839874344, −4.83194013786201499289753155223, −4.80273641950525008575800480041, −4.71767930993528865742955735291, −4.06295288384877729185177902367, −3.90237761685192721848804881707, −3.81129980392009084672953383623, −2.81402510949329931380701325419, −2.72597915358221971344266696841, −2.56027839443156978482030679057, −2.15182445368636131711030047043, −1.27028259800030546082233228921, −0.66479052213457971802586217344,
0.66479052213457971802586217344, 1.27028259800030546082233228921, 2.15182445368636131711030047043, 2.56027839443156978482030679057, 2.72597915358221971344266696841, 2.81402510949329931380701325419, 3.81129980392009084672953383623, 3.90237761685192721848804881707, 4.06295288384877729185177902367, 4.71767930993528865742955735291, 4.80273641950525008575800480041, 4.83194013786201499289753155223, 5.64198617118707104973839874344, 5.87874239834461993758233121628, 6.13298127990956816676196586426, 6.19016589326761072017604130012, 6.51574196723334306186197570090, 6.62749018033338325467222732199, 7.05596577721133797356519805652, 7.67711288871405901084678875089, 7.75002290152873825471492779088, 7.76599749664053792588305504743, 8.413903034769275654190331708903, 8.429955814812030758060907150571, 8.624492040485525580235641918852