| L(s) = 1 | + 2·2-s + 4-s + 2·7-s − 2·8-s + 9-s − 6·11-s + 10·13-s + 4·14-s − 4·16-s + 6·17-s + 2·18-s − 6·19-s − 12·22-s − 18·23-s + 20·26-s + 2·28-s − 2·29-s − 2·32-s + 12·34-s + 36-s − 14·37-s − 12·38-s + 36·41-s − 30·43-s − 6·44-s − 36·46-s − 12·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.755·7-s − 0.707·8-s + 1/3·9-s − 1.80·11-s + 2.77·13-s + 1.06·14-s − 16-s + 1.45·17-s + 0.471·18-s − 1.37·19-s − 2.55·22-s − 3.75·23-s + 3.92·26-s + 0.377·28-s − 0.371·29-s − 0.353·32-s + 2.05·34-s + 1/6·36-s − 2.30·37-s − 1.94·38-s + 5.62·41-s − 4.57·43-s − 0.904·44-s − 5.30·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.179576191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179576191\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 4 T^{3} + 67 T^{4} + 4 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 18 T + 180 T^{2} + 1296 T^{3} + 7139 T^{4} + 1296 p T^{5} + 180 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 22 T^{3} + 1252 T^{4} - 22 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 14 T + 85 T^{2} + 14 p T^{3} + 100 p T^{4} + 14 p^{2} T^{5} + 85 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T + 621 T^{2} - 6804 T^{3} + 51752 T^{4} - 6804 p T^{5} + 621 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 30 T + 460 T^{2} + 4800 T^{3} + 36651 T^{4} + 4800 p T^{5} + 460 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 170 T^{2} + 12411 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T + 16 T^{2} + 292 T^{3} - 4613 T^{4} + 292 p T^{5} + 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 14307 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 16 T + 207 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.51388147289340401254905336017, −6.20259266868804239162534777855, −6.00947239291683402568949377112, −5.86759757102522723551704142941, −5.68632840653515693347221698610, −5.23554281656643299450088570798, −5.21973842170854667271755696610, −5.21707547889817224781002011122, −5.03203317096278843118864203705, −4.47698057287811357499157176854, −4.24480508381831984261460304213, −4.18168770657686480966034297343, −3.87461574389933656592922320177, −3.75050353316708022148363427190, −3.45019214436369990884490863909, −3.35307841932607483930229376490, −3.33324269004847434731198677420, −2.46173035828381615471844561235, −2.31656302522527262189697006232, −2.21026839863276041520488937216, −2.12838362211561024869150728532, −1.45125904728160548808168282466, −1.27932508990451918092554064412, −0.869156596252811322291738366668, −0.14320750583377898379455311079,
0.14320750583377898379455311079, 0.869156596252811322291738366668, 1.27932508990451918092554064412, 1.45125904728160548808168282466, 2.12838362211561024869150728532, 2.21026839863276041520488937216, 2.31656302522527262189697006232, 2.46173035828381615471844561235, 3.33324269004847434731198677420, 3.35307841932607483930229376490, 3.45019214436369990884490863909, 3.75050353316708022148363427190, 3.87461574389933656592922320177, 4.18168770657686480966034297343, 4.24480508381831984261460304213, 4.47698057287811357499157176854, 5.03203317096278843118864203705, 5.21707547889817224781002011122, 5.21973842170854667271755696610, 5.23554281656643299450088570798, 5.68632840653515693347221698610, 5.86759757102522723551704142941, 6.00947239291683402568949377112, 6.20259266868804239162534777855, 6.51388147289340401254905336017
