| L(s) = 1 | − 4-s − 18·9-s − 12·11-s + 16-s − 128·19-s + 504·29-s + 80·31-s + 18·36-s − 900·41-s + 12·44-s − 98·49-s − 1.60e3·59-s − 856·61-s − 65·64-s + 1.90e3·71-s + 128·76-s + 1.14e3·79-s + 243·81-s − 732·89-s + 216·99-s + 2.29e3·101-s + 1.00e3·109-s − 504·116-s − 2.38e3·121-s − 80·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/8·4-s − 2/3·9-s − 0.328·11-s + 1/64·16-s − 1.54·19-s + 3.22·29-s + 0.463·31-s + 1/12·36-s − 3.42·41-s + 0.0411·44-s − 2/7·49-s − 3.54·59-s − 1.79·61-s − 0.126·64-s + 3.18·71-s + 0.193·76-s + 1.62·79-s + 1/3·81-s − 0.871·89-s + 0.219·99-s + 2.25·101-s + 0.878·109-s − 0.403·116-s − 1.79·121-s − 0.0579·124-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
| \(L(2)\) |
\(\approx\) |
\(0.5529216992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5529216992\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_4\times C_2$ | \( 1 + T^{2} + p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 6 T + 1246 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4556 T^{2} + 14317590 T^{4} - 4556 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 19520 T^{2} + 143530686 T^{4} - 19520 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 64 T + 6534 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 15704 T^{2} + 357132654 T^{4} - 15704 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 252 T + 56446 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 134924 T^{2} + 8546718390 T^{4} - 134924 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 181676 T^{2} + 16252611510 T^{4} - 181676 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 283436 T^{2} + 41632933734 T^{4} - 283436 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 14356 T^{2} + 44102355894 T^{4} + 14356 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 804 T + 380614 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 428 T + 425886 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 859676 T^{2} + 362036762934 T^{4} - 859676 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 977372 T^{2} + 539124333606 T^{4} - 977372 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 572 T + 901662 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 134924 T^{2} + 162125495574 T^{4} - 134924 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 366 T + 1156090 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1153244 T^{2} + 1289747845830 T^{4} - 1153244 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−7.46029527042246153351728887922, −7.08697797149362872071885649091, −6.90421590248875190456124920058, −6.47519899837809723402562832447, −6.37837933494822236795383319434, −6.32968178172018705885140431910, −6.15319120093791774961963025557, −5.75704740163529757987255810300, −5.21943972589373640909981174968, −5.10837873584692825721558446935, −4.83518221657139467560580866086, −4.80450666101873344190681166910, −4.47446653380665535120407296739, −4.01677790646131474900435915691, −3.86996060890320856476038129751, −3.40454739710177093488604277805, −3.00031539617151425531059670644, −2.94947668077506876579627308906, −2.77815353232561105059412950443, −2.07164878521730729187503599492, −1.87751376456170625084113361430, −1.65657257572099203224233433504, −0.963608566570048290128864327638, −0.67574010740331151210242140536, −0.12620848996950542813967775816,
0.12620848996950542813967775816, 0.67574010740331151210242140536, 0.963608566570048290128864327638, 1.65657257572099203224233433504, 1.87751376456170625084113361430, 2.07164878521730729187503599492, 2.77815353232561105059412950443, 2.94947668077506876579627308906, 3.00031539617151425531059670644, 3.40454739710177093488604277805, 3.86996060890320856476038129751, 4.01677790646131474900435915691, 4.47446653380665535120407296739, 4.80450666101873344190681166910, 4.83518221657139467560580866086, 5.10837873584692825721558446935, 5.21943972589373640909981174968, 5.75704740163529757987255810300, 6.15319120093791774961963025557, 6.32968178172018705885140431910, 6.37837933494822236795383319434, 6.47519899837809723402562832447, 6.90421590248875190456124920058, 7.08697797149362872071885649091, 7.46029527042246153351728887922
