| L(s) = 1 | − 2·4-s + 5·9-s + 4·11-s + 3·16-s − 12·19-s + 4·29-s + 34·31-s − 10·36-s + 20·41-s − 8·44-s + 17·49-s + 38·59-s − 6·61-s − 4·64-s + 8·71-s + 24·76-s − 14·79-s + 4·81-s + 12·89-s + 20·99-s + 22·101-s − 12·109-s − 8·116-s − 8·121-s − 68·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s + 5/3·9-s + 1.20·11-s + 3/4·16-s − 2.75·19-s + 0.742·29-s + 6.10·31-s − 5/3·36-s + 3.12·41-s − 1.20·44-s + 17/7·49-s + 4.94·59-s − 0.768·61-s − 1/2·64-s + 0.949·71-s + 2.75·76-s − 1.57·79-s + 4/9·81-s + 1.27·89-s + 2.01·99-s + 2.18·101-s − 1.14·109-s − 0.742·116-s − 0.727·121-s − 6.10·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.287965827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.287965827\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 7 p T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 17 T^{2} + 141 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 841 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 1505 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 29 T^{2} + 1005 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 17 T + 131 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 4974 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 161 T^{2} + 10149 T^{4} - 161 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5030 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 59 T^{2} + 4769 T^{4} + 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 19 T + 205 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 3 T + 121 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 27 T^{2} + 10681 T^{4} + 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 7 T + 141 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 11166 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 11521 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.89600837710128038279716916555, −6.56772075218612425649020988405, −6.28809117995147038902524273314, −6.21772236497891081578530631090, −6.06757015189466619920515615657, −5.77685282411163153341699446803, −5.51029168995967170844649015246, −5.19447288724455406323611802440, −4.84119530760227908915465267476, −4.60099350056560205776620827954, −4.42803433230484454317659436846, −4.31935601468584348403480455592, −4.30985046753736545249123150432, −3.88008541843714420732471338040, −3.87780207179798395143191010039, −3.52855398452850451515124902677, −2.88057096306761775705235458345, −2.84628327092854890665428303424, −2.33639587063362683445267341666, −2.23690256114523326514541631464, −2.19773890167835570569176934007, −1.22269177486588105291891970859, −1.07662233890556900058323718633, −0.968309749179320741965891951673, −0.58035298103125132904273288199,
0.58035298103125132904273288199, 0.968309749179320741965891951673, 1.07662233890556900058323718633, 1.22269177486588105291891970859, 2.19773890167835570569176934007, 2.23690256114523326514541631464, 2.33639587063362683445267341666, 2.84628327092854890665428303424, 2.88057096306761775705235458345, 3.52855398452850451515124902677, 3.87780207179798395143191010039, 3.88008541843714420732471338040, 4.30985046753736545249123150432, 4.31935601468584348403480455592, 4.42803433230484454317659436846, 4.60099350056560205776620827954, 4.84119530760227908915465267476, 5.19447288724455406323611802440, 5.51029168995967170844649015246, 5.77685282411163153341699446803, 6.06757015189466619920515615657, 6.21772236497891081578530631090, 6.28809117995147038902524273314, 6.56772075218612425649020988405, 6.89600837710128038279716916555
