| L(s) = 1 | − 2·4-s − 2·9-s + 2·11-s + 3·16-s + 6·19-s − 12·29-s + 4·31-s + 4·36-s + 12·41-s − 4·44-s + 19·49-s + 36·59-s + 24·61-s − 4·64-s − 6·71-s − 12·76-s + 42·79-s + 3·81-s + 30·89-s − 4·99-s + 6·101-s + 28·109-s + 24·116-s − 33·121-s − 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 0.603·11-s + 3/4·16-s + 1.37·19-s − 2.22·29-s + 0.718·31-s + 2/3·36-s + 1.87·41-s − 0.603·44-s + 19/7·49-s + 4.68·59-s + 3.07·61-s − 1/2·64-s − 0.712·71-s − 1.37·76-s + 4.72·79-s + 1/3·81-s + 3.17·89-s − 0.402·99-s + 0.597·101-s + 2.68·109-s + 2.22·116-s − 3·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.460033390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.460033390\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 19 T^{2} + 184 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 766 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 59 T^{2} + 1720 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5806 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 53 T^{2} + 1744 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 152 T^{2} + 10126 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 23 T^{2} + 4216 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 4254 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 79 T^{2} + 12112 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 21 T + 230 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7830 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 15 T + 196 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.86849769472132214116281617817, −5.51142217494776852550288840647, −5.46939516065669946944520553818, −5.25549167855958251282433018805, −5.19643992305739386259010314904, −5.04121553231473481601930658221, −4.72189350679260254605982719829, −4.39734108172500751164141073311, −4.12487037964941697116975643156, −4.10570864890589416960167320056, −3.93805618749383463815397868393, −3.68828068135981655495204589273, −3.43426961638120682080913853913, −3.30929073167262113917873258344, −3.26092895629926949272166384959, −2.67415892373530616426787958422, −2.48778996783538632993947573265, −2.22282443393755612369413455493, −2.16597438457862505212061249361, −1.96165774967665834184026430781, −1.51204612730515605206824358824, −0.959054634563096058020569026454, −0.69339981519109724514374122871, −0.66502963529881758785955748632, −0.63357346765610082826194222411,
0.63357346765610082826194222411, 0.66502963529881758785955748632, 0.69339981519109724514374122871, 0.959054634563096058020569026454, 1.51204612730515605206824358824, 1.96165774967665834184026430781, 2.16597438457862505212061249361, 2.22282443393755612369413455493, 2.48778996783538632993947573265, 2.67415892373530616426787958422, 3.26092895629926949272166384959, 3.30929073167262113917873258344, 3.43426961638120682080913853913, 3.68828068135981655495204589273, 3.93805618749383463815397868393, 4.10570864890589416960167320056, 4.12487037964941697116975643156, 4.39734108172500751164141073311, 4.72189350679260254605982719829, 5.04121553231473481601930658221, 5.19643992305739386259010314904, 5.25549167855958251282433018805, 5.46939516065669946944520553818, 5.51142217494776852550288840647, 5.86849769472132214116281617817
