| L(s) = 1 | + 2·4-s + 4·5-s + 6·9-s + 4·13-s + 3·16-s + 8·20-s − 8·23-s − 10·25-s + 12·36-s + 24·45-s − 12·49-s + 8·52-s + 4·53-s + 8·59-s + 12·64-s + 16·65-s + 24·71-s + 12·80-s + 17·81-s + 8·83-s − 16·92-s − 20·100-s − 8·103-s − 48·107-s − 28·109-s − 32·115-s + 24·117-s + ⋯ |
| L(s) = 1 | + 4-s + 1.78·5-s + 2·9-s + 1.10·13-s + 3/4·16-s + 1.78·20-s − 1.66·23-s − 2·25-s + 2·36-s + 3.57·45-s − 1.71·49-s + 1.10·52-s + 0.549·53-s + 1.04·59-s + 3/2·64-s + 1.98·65-s + 2.84·71-s + 1.34·80-s + 17/9·81-s + 0.878·83-s − 1.66·92-s − 2·100-s − 0.788·103-s − 4.64·107-s − 2.68·109-s − 2.98·115-s + 2.21·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.037093135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.037093135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 29 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 595 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T^{2} + 131 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T^{2} - 1210 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 118 T^{2} + 6979 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 150 T^{2} + 9971 T^{4} - 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 19414 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4$ | \( ( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 310 T^{2} + 36499 T^{4} - 310 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 19334 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 25446 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.25445764760947490713352480536, −6.94069113933645045785152950480, −6.77608095644138810887472453383, −6.65611525397985221952073520646, −6.49320206463116236873727458648, −6.10235573169927058960045841122, −5.94659844671640494982075655746, −5.82236966444007733438604789254, −5.49506598388853544238877724714, −5.45111779888179061077947235226, −5.06867995510784423562311997936, −4.71637901956536614673745785569, −4.42006798095713443909397669926, −3.98926573870603428691897100429, −3.98357917832470431981663563671, −3.70162289364610289297689955157, −3.52542532797285951597610386244, −3.10264437230421786189216456328, −2.40105092694515958317126692962, −2.31551975608232533267039205087, −2.23185104576905636266353422527, −1.68558100572780546595009048676, −1.48273327199274240323310656581, −1.44667857738918146319165540035, −0.57153214305092546042254282853,
0.57153214305092546042254282853, 1.44667857738918146319165540035, 1.48273327199274240323310656581, 1.68558100572780546595009048676, 2.23185104576905636266353422527, 2.31551975608232533267039205087, 2.40105092694515958317126692962, 3.10264437230421786189216456328, 3.52542532797285951597610386244, 3.70162289364610289297689955157, 3.98357917832470431981663563671, 3.98926573870603428691897100429, 4.42006798095713443909397669926, 4.71637901956536614673745785569, 5.06867995510784423562311997936, 5.45111779888179061077947235226, 5.49506598388853544238877724714, 5.82236966444007733438604789254, 5.94659844671640494982075655746, 6.10235573169927058960045841122, 6.49320206463116236873727458648, 6.65611525397985221952073520646, 6.77608095644138810887472453383, 6.94069113933645045785152950480, 7.25445764760947490713352480536
