| L(s) = 1 | − 2·5-s − 7-s + 11-s + 13-s − 4·17-s + 4·19-s − 5·23-s + 25-s + 29-s − 5·31-s + 2·35-s + 24·37-s − 12·41-s − 2·43-s + 13·47-s + 6·49-s + 6·53-s − 2·55-s − 10·59-s − 14·61-s − 2·65-s − 10·67-s − 6·71-s + 36·73-s − 77-s − 6·79-s + 20·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.301·11-s + 0.277·13-s − 0.970·17-s + 0.917·19-s − 1.04·23-s + 1/5·25-s + 0.185·29-s − 0.898·31-s + 0.338·35-s + 3.94·37-s − 1.87·41-s − 0.304·43-s + 1.89·47-s + 6/7·49-s + 0.824·53-s − 0.269·55-s − 1.30·59-s − 1.79·61-s − 0.248·65-s − 1.22·67-s − 0.712·71-s + 4.21·73-s − 0.113·77-s − 0.675·79-s + 2.19·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079834172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079834172\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + T - 5 T^{2} - 8 T^{3} - 20 T^{4} - 8 p T^{5} - 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - T - 13 T^{2} + 8 T^{3} + 64 T^{4} + 8 p T^{5} - 13 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - T - 17 T^{2} + 8 T^{3} + 142 T^{4} + 8 p T^{5} - 17 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 10 p T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - T - 49 T^{2} + 8 T^{3} + 1630 T^{4} + 8 p T^{5} - 49 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 59 T^{2} + 36 T^{3} - 360 T^{4} + 36 p T^{5} + 59 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 50 T^{2} - 64 T^{3} + 895 T^{4} - 64 p T^{5} - 50 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 13 T + 41 T^{2} - 442 T^{3} + 6232 T^{4} - 442 p T^{5} + 41 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 14 T + 58 T^{2} + 224 T^{3} + 3367 T^{4} + 224 p T^{5} + 58 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 26 T^{2} - 80 T^{3} + 4687 T^{4} - 80 p T^{5} - 26 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 136 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 98 T^{2} - 144 T^{3} + 8871 T^{4} - 144 p T^{5} - 98 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 10 T + 17 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 158 T^{2} - 64 T^{3} + 16447 T^{4} - 64 p T^{5} - 158 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.14974508472976308646257103100, −6.00619008132579742250560268770, −5.87892676608931710184553514249, −5.49456300485960039319896367956, −5.33887721521427467926497644227, −5.02597220215962631548597591623, −4.96549199622180954284953496943, −4.60180017813429527547297792347, −4.45766165119970375043019896783, −4.30111287031308873741312825834, −3.96321745295237657145517359799, −3.90623230718211332041477573684, −3.68397784113886277262093069387, −3.45299238922007081639274781026, −3.23780986880018604915057791034, −2.98370884978573665668090115290, −2.52181075498150041577512934964, −2.49270678857820566422188232403, −2.33720013513083589688524045294, −2.01957698863090242111830319791, −1.47004096493357825723198046937, −1.34060094055127216178423033709, −1.01573740121230806040181460743, −0.61265783893763797875023291018, −0.19694815690425273520224697858,
0.19694815690425273520224697858, 0.61265783893763797875023291018, 1.01573740121230806040181460743, 1.34060094055127216178423033709, 1.47004096493357825723198046937, 2.01957698863090242111830319791, 2.33720013513083589688524045294, 2.49270678857820566422188232403, 2.52181075498150041577512934964, 2.98370884978573665668090115290, 3.23780986880018604915057791034, 3.45299238922007081639274781026, 3.68397784113886277262093069387, 3.90623230718211332041477573684, 3.96321745295237657145517359799, 4.30111287031308873741312825834, 4.45766165119970375043019896783, 4.60180017813429527547297792347, 4.96549199622180954284953496943, 5.02597220215962631548597591623, 5.33887721521427467926497644227, 5.49456300485960039319896367956, 5.87892676608931710184553514249, 6.00619008132579742250560268770, 6.14974508472976308646257103100
