| L(s) = 1 | + 19·4-s − 18·9-s − 62·11-s + 181·16-s + 112·19-s + 124·29-s − 270·31-s − 342·36-s + 470·41-s − 1.17e3·44-s − 98·49-s + 882·59-s − 446·61-s + 1.23e3·64-s + 1.23e3·71-s + 2.12e3·76-s + 854·79-s + 243·81-s − 932·89-s + 1.11e3·99-s − 868·101-s + 3.59e3·109-s + 2.35e3·116-s − 2.70e3·121-s − 5.13e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 19/8·4-s − 2/3·9-s − 1.69·11-s + 2.82·16-s + 1.35·19-s + 0.794·29-s − 1.56·31-s − 1.58·36-s + 1.79·41-s − 4.03·44-s − 2/7·49-s + 1.94·59-s − 0.936·61-s + 2.41·64-s + 2.06·71-s + 3.21·76-s + 1.21·79-s + 1/3·81-s − 1.11·89-s + 1.13·99-s − 0.855·101-s + 3.15·109-s + 1.88·116-s − 2.03·121-s − 3.71·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.370721329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370721329\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - 19 T^{2} + 45 p^{2} T^{4} - 19 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 31 T + 2796 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5571 T^{2} + 15544460 T^{4} - 5571 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10335 T^{2} + 55642016 T^{4} - 10335 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 56 T + 1174 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 11514 T^{2} + 171279659 T^{4} - 11514 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 62 T + 19751 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 135 T + 63420 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 193159 T^{2} + 14439461800 T^{4} - 193159 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 235 T + 143042 T^{2} - 235 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 33774 T^{2} + 3686087315 T^{4} + 33774 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 337916 T^{2} + 49344837574 T^{4} - 337916 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 517079 T^{2} + 110424015304 T^{4} - 517079 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 441 T + 273734 T^{2} - 441 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 223 T + 149646 T^{2} + 223 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 405131 T^{2} + 135876758064 T^{4} - 405131 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 619 T + 576906 T^{2} - 619 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1454332 T^{2} + 829074972646 T^{4} - 1454332 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 427 T + 986572 T^{2} - 427 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 25081 T^{2} - 503757220916 T^{4} + 25081 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 466 T + 306170 T^{2} + 466 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 576836 T^{2} + 1686814161670 T^{4} + 576836 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−7.51481824335080407132210729681, −6.92499393345781832257708128793, −6.88807390509713953975863721367, −6.88395025011908049600612009023, −6.67812544748660144602872388789, −5.93589168810869427689729692270, −5.93263983812201414376527455143, −5.81761910568531620050127098119, −5.63790619207690968559602711578, −5.25918058901822873795821044937, −4.95847608809309876419895816283, −4.71225728142517681928771410820, −4.52524891133272392457023782904, −3.89311467075332569135161413723, −3.41908978691637535910624205904, −3.41091311709315535688027812200, −3.33464766445126808867989313296, −2.50740341891656517050477106539, −2.41835982132778132157490361484, −2.41124529685428203914162521764, −2.25025037778850724628986485304, −1.45469342846911195459752545223, −1.20949281736065229965440745536, −0.838207313846908625784818746338, −0.17998808093967275412602468942,
0.17998808093967275412602468942, 0.838207313846908625784818746338, 1.20949281736065229965440745536, 1.45469342846911195459752545223, 2.25025037778850724628986485304, 2.41124529685428203914162521764, 2.41835982132778132157490361484, 2.50740341891656517050477106539, 3.33464766445126808867989313296, 3.41091311709315535688027812200, 3.41908978691637535910624205904, 3.89311467075332569135161413723, 4.52524891133272392457023782904, 4.71225728142517681928771410820, 4.95847608809309876419895816283, 5.25918058901822873795821044937, 5.63790619207690968559602711578, 5.81761910568531620050127098119, 5.93263983812201414376527455143, 5.93589168810869427689729692270, 6.67812544748660144602872388789, 6.88395025011908049600612009023, 6.88807390509713953975863721367, 6.92499393345781832257708128793, 7.51481824335080407132210729681
