| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s − 2·8-s + 9-s − 6·11-s − 2·12-s + 6·13-s − 4·16-s + 8·17-s + 2·18-s − 12·22-s − 2·23-s + 4·24-s + 12·26-s + 2·27-s + 4·29-s + 24·31-s − 2·32-s + 12·33-s + 16·34-s + 36-s + 2·37-s − 12·39-s + 4·43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s − 0.707·8-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 1.66·13-s − 16-s + 1.94·17-s + 0.471·18-s − 2.55·22-s − 0.417·23-s + 0.816·24-s + 2.35·26-s + 0.384·27-s + 0.742·29-s + 4.31·31-s − 0.353·32-s + 2.08·33-s + 2.74·34-s + 1/6·36-s + 0.328·37-s − 1.92·39-s + 0.609·43-s − 0.904·44-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
| \(L(1)\) |
\(\approx\) |
\(5.664518930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.664518930\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 - 4 T^{2} - 33 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 24 T^{2} - 48 T^{3} + 223 T^{4} - 48 p T^{5} + 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 28 T^{2} + 423 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 33 T^{2} - 18 T^{3} + 748 T^{4} - 18 p T^{5} - 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 6 T^{2} + 144 T^{3} - 821 T^{4} + 144 p T^{5} - 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 88 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T + 19 T^{2} + 178 T^{3} - 1292 T^{4} + 178 p T^{5} + 19 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 8 T^{2} - 1617 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T - 64 T^{2} + 24 T^{3} + 3863 T^{4} + 24 p T^{5} - 64 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 112 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 10 T + 43 T^{2} - 650 T^{3} - 6572 T^{4} - 650 p T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 22 T + 231 T^{2} - 2442 T^{3} + 24604 T^{4} - 2442 p T^{5} + 231 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 131 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 120 T^{2} - 48 T^{3} + 20239 T^{4} - 48 p T^{5} - 120 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 22 T + 209 T^{2} - 1782 T^{3} + 19268 T^{4} - 1782 p T^{5} + 209 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.39320162844881283998994124403, −6.24185532299977446302719383242, −5.97650479436614851950355463044, −5.83207094531690615430463687370, −5.78644303574987755314720770493, −5.45188041536133469734599887845, −5.20072595442240572271621546539, −5.06309033030069985916438143312, −4.70247493813329533911295912825, −4.70152514332785901390009478406, −4.61939373899715409828891717652, −4.24527906370747133682141271236, −4.08383994305482555174555166970, −3.44630082565955156361948680160, −3.40852924101635128549632369989, −3.28332978642052315146386622603, −3.18260991748080701793626898977, −2.75785635420347541079938437872, −2.59313621490007519456878056139, −2.14183770038641545691964656248, −1.89279160720210128312824440071, −1.43737358906594038769822332553, −0.900191078478012616506421748714, −0.78281978188852964691869145124, −0.48594750139780492993513293480,
0.48594750139780492993513293480, 0.78281978188852964691869145124, 0.900191078478012616506421748714, 1.43737358906594038769822332553, 1.89279160720210128312824440071, 2.14183770038641545691964656248, 2.59313621490007519456878056139, 2.75785635420347541079938437872, 3.18260991748080701793626898977, 3.28332978642052315146386622603, 3.40852924101635128549632369989, 3.44630082565955156361948680160, 4.08383994305482555174555166970, 4.24527906370747133682141271236, 4.61939373899715409828891717652, 4.70152514332785901390009478406, 4.70247493813329533911295912825, 5.06309033030069985916438143312, 5.20072595442240572271621546539, 5.45188041536133469734599887845, 5.78644303574987755314720770493, 5.83207094531690615430463687370, 5.97650479436614851950355463044, 6.24185532299977446302719383242, 6.39320162844881283998994124403
