| L(s) = 1 | + 2·2-s + 4-s − 2·5-s − 2·8-s − 9-s − 4·10-s − 2·11-s + 20·13-s − 4·16-s − 12·17-s − 2·18-s + 6·19-s − 2·20-s − 4·22-s + 2·23-s + 4·25-s + 40·26-s + 4·29-s + 8·31-s − 2·32-s − 24·34-s − 36-s − 2·37-s + 12·38-s + 4·40-s + 12·41-s − 16·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.707·8-s − 1/3·9-s − 1.26·10-s − 0.603·11-s + 5.54·13-s − 16-s − 2.91·17-s − 0.471·18-s + 1.37·19-s − 0.447·20-s − 0.852·22-s + 0.417·23-s + 4/5·25-s + 7.84·26-s + 0.742·29-s + 1.43·31-s − 0.353·32-s − 4.11·34-s − 1/6·36-s − 0.328·37-s + 1.94·38-s + 0.632·40-s + 1.87·41-s − 2.43·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.297646822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.297646822\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 4 T^{2} - 12 T^{3} + 555 T^{4} - 12 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 36 T^{2} + 12 T^{3} + 979 T^{4} + 12 p T^{5} - 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 12 p T^{5} - 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 126 T^{2} + 576 T^{3} + 2659 T^{4} + 576 p T^{5} + 126 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2 T - 96 T^{2} + 12 T^{3} + 6979 T^{4} + 12 p T^{5} - 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 99 T^{2} - 12 T^{3} + 8800 T^{4} - 12 p T^{5} - 99 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 954 p T^{5} + 149 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 23 T^{2} + 376 T^{3} - 1208 T^{4} + 376 p T^{5} - 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 14 T + 184 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 12 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 768 p T^{5} - 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.430245649503273250634322470650, −9.169116142552141687309255412640, −9.034082311170186691341034692542, −8.550679611979928545023305608530, −8.346903938502269848788683500109, −8.134117974988046124052436441104, −8.021986884520513601419562775696, −7.968035872748558359004354378445, −6.88983008761910233471025359542, −6.69797574880814778690093801995, −6.53744348665975916969101268034, −6.50950775462488720817289925770, −5.96478698770643686587908090181, −5.93403474009816346402816364224, −5.26612020791751810280609560819, −5.02323960448957484868037720839, −4.70840771771047264601916013173, −4.41205116391757646211617602553, −3.79638292825034123679365534386, −3.73725266090806095837061049493, −3.57450956170679545052740211294, −2.96298327526848233382819988489, −2.77152966487750449609004297313, −1.68186501093571855182183849520, −1.08928941804182991840182327617,
1.08928941804182991840182327617, 1.68186501093571855182183849520, 2.77152966487750449609004297313, 2.96298327526848233382819988489, 3.57450956170679545052740211294, 3.73725266090806095837061049493, 3.79638292825034123679365534386, 4.41205116391757646211617602553, 4.70840771771047264601916013173, 5.02323960448957484868037720839, 5.26612020791751810280609560819, 5.93403474009816346402816364224, 5.96478698770643686587908090181, 6.50950775462488720817289925770, 6.53744348665975916969101268034, 6.69797574880814778690093801995, 6.88983008761910233471025359542, 7.968035872748558359004354378445, 8.021986884520513601419562775696, 8.134117974988046124052436441104, 8.346903938502269848788683500109, 8.550679611979928545023305608530, 9.034082311170186691341034692542, 9.169116142552141687309255412640, 9.430245649503273250634322470650
