| L(s) = 1 | + 64·4-s + 308·7-s + 4.44e3·11-s + 3.07e3·16-s − 4.05e4·23-s + 1.09e4·25-s + 1.97e4·28-s + 1.82e4·29-s − 2.31e4·37-s − 4.46e4·43-s + 2.84e5·44-s − 1.07e5·49-s − 2.48e5·53-s + 1.31e5·64-s − 4.34e5·67-s + 4.51e5·71-s + 1.36e6·77-s + 2.09e6·79-s − 2.59e6·92-s + 6.97e5·100-s − 6.34e5·107-s + 5.82e6·109-s + 9.46e5·112-s − 2.93e6·113-s + 1.16e6·116-s + 5.24e6·121-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s + 0.897·7-s + 3.33·11-s + 3/4·16-s − 3.33·23-s + 0.697·25-s + 0.897·28-s + 0.748·29-s − 0.457·37-s − 0.562·43-s + 3.33·44-s − 0.915·49-s − 1.66·53-s + 1/2·64-s − 1.44·67-s + 1.26·71-s + 2.99·77-s + 4.24·79-s − 3.33·92-s + 0.697·100-s − 0.517·107-s + 4.49·109-s + 0.673·112-s − 2.03·113-s + 0.748·116-s + 2.96·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(11.89206106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.89206106\) |
| \(L(4)\) |
|
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 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 - 44 p T + 4134 p^{2} T^{2} - 44 p^{7} T^{3} + p^{12} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 436 p^{2} T^{2} + 412902 p^{4} T^{4} - 436 p^{14} T^{6} + p^{24} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 2220 T + 4770614 T^{2} - 2220 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 3940180 T^{2} + 17707018928070 T^{4} - 3940180 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 56335492 T^{2} + 1681094907776646 T^{4} - 56335492 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 122207956 T^{2} + 7623942006639558 T^{4} - 122207956 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 20292 T + 396782822 T^{2} + 20292 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 9132 T - 168260074 T^{2} - 9132 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1275081988 T^{2} + 1557093819506538246 T^{4} - 1275081988 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 11596 T + 3283526454 T^{2} + 11596 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 9929192068 T^{2} + 52045566850295970246 T^{4} - 9929192068 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 22348 T + 2121261174 T^{2} + 22348 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 22996383364 T^{2} + \)\(27\!\cdots\!18\)\( T^{4} - 22996383364 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 124308 T + 44342337206 T^{2} + 124308 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 11871974636 T^{2} + \)\(33\!\cdots\!86\)\( T^{4} + 11871974636 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 84307090516 T^{2} + \)\(41\!\cdots\!58\)\( T^{4} - 84307090516 p^{12} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 217388 T + 192725823126 T^{2} + 217388 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 225804 T + 160756795334 T^{2} - 225804 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 377326906756 T^{2} + \)\(71\!\cdots\!98\)\( T^{4} - 377326906756 p^{12} T^{6} + p^{24} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 1046452 T + 754787202918 T^{2} - 1046452 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 828919590868 T^{2} + \)\(35\!\cdots\!70\)\( T^{4} - 828919590868 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 1729674521860 T^{2} + \)\(12\!\cdots\!30\)\( T^{4} - 1729674521860 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 3289276805380 T^{2} + \)\(40\!\cdots\!70\)\( T^{4} - 3289276805380 p^{12} T^{6} + p^{24} T^{8} \) |
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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
−8.574501623127867260061823461191, −8.405571909526670719049240702219, −7.958218603881286631972880526090, −7.81327997085542631886543339073, −7.63641226690409251845421658663, −7.07623092606161981621636738501, −6.78504349610156983966551618145, −6.46543324374571441906437738523, −6.35361847434428120789235650637, −6.15192591960279909391670306782, −5.94878456896717928746531568826, −5.17263212009447503262047553138, −5.06921031498736916572184311828, −4.61613735270410227379909318751, −4.13421135108419735858837300819, −4.00690286353952307825907238070, −3.60664323440448532356149872075, −3.37591819167105495701621906131, −2.81102252157328293074285068401, −2.25433374179099642813636234489, −1.75363741766441616576829651434, −1.57362407414439695325193588563, −1.56397152098269193094408580336, −0.68383304152371832741143147218, −0.52299981241038279621631283301,
0.52299981241038279621631283301, 0.68383304152371832741143147218, 1.56397152098269193094408580336, 1.57362407414439695325193588563, 1.75363741766441616576829651434, 2.25433374179099642813636234489, 2.81102252157328293074285068401, 3.37591819167105495701621906131, 3.60664323440448532356149872075, 4.00690286353952307825907238070, 4.13421135108419735858837300819, 4.61613735270410227379909318751, 5.06921031498736916572184311828, 5.17263212009447503262047553138, 5.94878456896717928746531568826, 6.15192591960279909391670306782, 6.35361847434428120789235650637, 6.46543324374571441906437738523, 6.78504349610156983966551618145, 7.07623092606161981621636738501, 7.63641226690409251845421658663, 7.81327997085542631886543339073, 7.958218603881286631972880526090, 8.405571909526670719049240702219, 8.574501623127867260061823461191
