| L(s) = 1 | − 2·2-s + 2·4-s − 12·7-s − 4·8-s + 24·14-s + 8·16-s − 8·17-s − 16·23-s − 4·25-s − 24·28-s − 20·31-s − 8·32-s + 16·34-s − 8·41-s + 32·46-s − 20·47-s + 68·49-s + 8·50-s + 48·56-s + 40·62-s + 8·64-s − 16·68-s + 12·71-s − 16·73-s + 8·79-s + 16·82-s − 32·92-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 4.53·7-s − 1.41·8-s + 6.41·14-s + 2·16-s − 1.94·17-s − 3.33·23-s − 4/5·25-s − 4.53·28-s − 3.59·31-s − 1.41·32-s + 2.74·34-s − 1.24·41-s + 4.71·46-s − 2.91·47-s + 68/7·49-s + 1.13·50-s + 6.41·56-s + 5.08·62-s + 64-s − 1.94·68-s + 1.42·71-s − 1.87·73-s + 0.900·79-s + 1.76·82-s − 3.33·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 1866 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8618 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7734 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 25866 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 276 T^{2} + 32522 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.020941794939133584860518192864, −7.18706617342979307753758092314, −7.12450220630040367648499691281, −6.99854600532153647017091440480, −6.72006370074289614483089716787, −6.63784586481738135130986254858, −6.30362524727238835065348539218, −6.21000194626873928727063218920, −5.97233738629684496505553764677, −5.89668617770242784064811445644, −5.61637003307684074460220913949, −5.28293202589334371685282680310, −5.01435696178616976656579535507, −4.55837487039604919016566799370, −4.07346674060573431614367093117, −3.88022216881596183244909708408, −3.79463603611424123603995423590, −3.55815558896606981189675044694, −3.19448093856665652571622750036, −3.09355066450153156380876986634, −2.85657727637471904963457577851, −2.23791321779624822399883873520, −2.06517492412871521886346786147, −1.93801626642843690496073000937, −1.24190027439206878245471091921, 0, 0, 0, 0,
1.24190027439206878245471091921, 1.93801626642843690496073000937, 2.06517492412871521886346786147, 2.23791321779624822399883873520, 2.85657727637471904963457577851, 3.09355066450153156380876986634, 3.19448093856665652571622750036, 3.55815558896606981189675044694, 3.79463603611424123603995423590, 3.88022216881596183244909708408, 4.07346674060573431614367093117, 4.55837487039604919016566799370, 5.01435696178616976656579535507, 5.28293202589334371685282680310, 5.61637003307684074460220913949, 5.89668617770242784064811445644, 5.97233738629684496505553764677, 6.21000194626873928727063218920, 6.30362524727238835065348539218, 6.63784586481738135130986254858, 6.72006370074289614483089716787, 6.99854600532153647017091440480, 7.12450220630040367648499691281, 7.18706617342979307753758092314, 8.020941794939133584860518192864
