| L(s) = 1 | + 32·4-s + 410·7-s + 4.08e3·13-s − 6.00e3·19-s − 3.00e4·25-s + 1.31e4·28-s + 6.99e4·31-s − 2.30e5·37-s + 1.25e5·43-s + 2.77e5·49-s + 1.30e5·52-s + 1.22e5·61-s − 3.27e4·64-s − 1.35e5·67-s + 1.69e6·73-s − 1.92e5·76-s + 1.41e6·79-s + 1.67e6·91-s − 1.05e6·97-s − 9.63e5·100-s + 5.70e5·103-s + 8.85e6·109-s − 2.43e6·121-s + 2.23e6·124-s + 127-s + 131-s − 2.46e6·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.19·7-s + 1.85·13-s − 0.875·19-s − 1.92·25-s + 0.597·28-s + 2.34·31-s − 4.55·37-s + 1.57·43-s + 2.35·49-s + 0.928·52-s + 0.540·61-s − 1/8·64-s − 0.450·67-s + 4.35·73-s − 0.437·76-s + 2.87·79-s + 2.22·91-s − 1.15·97-s − 0.963·100-s + 0.522·103-s + 6.83·109-s − 1.37·121-s + 1.17·124-s − 1.04·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(8.893230367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.893230367\) |
| \(L(4)\) |
|
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 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 30098 T^{2} + 661748979 T^{4} + 30098 p^{12} T^{6} + p^{24} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 205 T - 75624 T^{2} - 205 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 2436050 T^{2} + 2795911225779 T^{4} + 2436050 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 157 p T - 3912 p^{2} T^{2} - 157 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 19749310 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 79 p T + p^{6} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 245751266 T^{2} + 38479060308582435 T^{4} + 245751266 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 380661554 T^{2} - 208911564511774125 T^{4} + 380661554 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 34990 T + 336796419 T^{2} - 34990 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 57625 T + p^{6} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 8546751646 T^{2} + 50483473398037523235 T^{4} - 8546751646 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 62566 T - 2406858693 T^{2} - 62566 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 18865237826 T^{2} + \)\(23\!\cdots\!35\)\( T^{4} + 18865237826 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 38371642706 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 53489235310 T^{2} + \)\(10\!\cdots\!19\)\( T^{4} - 53489235310 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 61297 T - 47763052152 T^{2} - 61297 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 67691 T - 85876310688 T^{2} + 67691 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 3414318046 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 423983 T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 707533 T + 257515490568 T^{2} - 707533 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 87355170574 T^{2} - \)\(99\!\cdots\!85\)\( T^{4} - 87355170574 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 548198208290 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 526151 T - 556137130128 T^{2} + 526151 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−8.213591273346551986322033029412, −8.051332853582129432717594961206, −7.88296931412457512915535770948, −7.46692523733391224029657346008, −7.08728345148794629905124755075, −6.79479001759124514851492605792, −6.61796700151419464634557335735, −6.37951479169710624735764366172, −5.81610949614541924458306271498, −5.70300270716701104192007447308, −5.60161403806600794908995456359, −4.93069333726196727619013610646, −4.82027391080730171364426587485, −4.38894370929594914401737786823, −4.08830520441828417752229253542, −3.64287154508504065228150373495, −3.38267251463268443258639927200, −3.32182845732779314006088055195, −2.36233648232392568331749008188, −2.12096955723894106122197438597, −2.00748743883235904696039615780, −1.64867296824593857901583009458, −0.948238583198703305343084625756, −0.817596573994817419314402719903, −0.39874791575154027116361606053,
0.39874791575154027116361606053, 0.817596573994817419314402719903, 0.948238583198703305343084625756, 1.64867296824593857901583009458, 2.00748743883235904696039615780, 2.12096955723894106122197438597, 2.36233648232392568331749008188, 3.32182845732779314006088055195, 3.38267251463268443258639927200, 3.64287154508504065228150373495, 4.08830520441828417752229253542, 4.38894370929594914401737786823, 4.82027391080730171364426587485, 4.93069333726196727619013610646, 5.60161403806600794908995456359, 5.70300270716701104192007447308, 5.81610949614541924458306271498, 6.37951479169710624735764366172, 6.61796700151419464634557335735, 6.79479001759124514851492605792, 7.08728345148794629905124755075, 7.46692523733391224029657346008, 7.88296931412457512915535770948, 8.051332853582129432717594961206, 8.213591273346551986322033029412
