| L(s) = 1 | + 2·2-s + 4-s − 6·7-s − 2·8-s + 9-s − 6·11-s − 14·13-s − 12·14-s − 4·16-s + 2·18-s − 6·19-s − 12·22-s − 12·23-s − 28·26-s − 6·28-s − 4·29-s − 2·32-s + 36-s + 2·37-s − 12·38-s − 6·44-s − 24·46-s − 12·47-s + 23·49-s − 14·52-s + 12·56-s − 8·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 2.26·7-s − 0.707·8-s + 1/3·9-s − 1.80·11-s − 3.88·13-s − 3.20·14-s − 16-s + 0.471·18-s − 1.37·19-s − 2.55·22-s − 2.50·23-s − 5.49·26-s − 1.13·28-s − 0.742·29-s − 0.353·32-s + 1/6·36-s + 0.328·37-s − 1.94·38-s − 0.904·44-s − 3.53·46-s − 1.75·47-s + 23/7·49-s − 1.94·52-s + 1.60·56-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 126 T^{3} + 452 T^{4} + 126 p T^{5} + 3 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 37 T^{2} + 150 T^{3} + 492 T^{4} + 150 p T^{5} + 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 34 T^{2} - 32 T^{3} + 1195 T^{4} - 32 p T^{5} - 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 1254 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T - 23 T^{2} + 94 T^{3} - 788 T^{4} + 94 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 66 T^{2} + 2675 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^3$ | \( 1 + 50 T^{2} + 651 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 198 T^{2} + 15371 T^{4} - 198 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 114 T^{2} - 792 T^{3} + 3707 T^{4} - 792 p T^{5} + 114 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 62 T^{2} - 32 T^{3} + 8251 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T - 110 T^{2} - 32 T^{3} + 10315 T^{4} - 32 p T^{5} - 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 24 T + 346 T^{2} + 3696 T^{3} + 32307 T^{4} + 3696 p T^{5} + 346 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 42 T + 909 T^{2} - 13482 T^{3} + 147452 T^{4} - 13482 p T^{5} + 909 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 74 T^{2} + 416 T^{3} - 2861 T^{4} + 416 p T^{5} - 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.96154377683027520326127900211, −6.55258345994619908649293073884, −6.44113201758756424590034046968, −6.38811075358275037106605262539, −6.01038533008779990631292437217, −5.82287036279973228969756087985, −5.73734517382288182070797573711, −5.26514233848340792891083683181, −5.21583453749440559231278847871, −5.18249385517977865122126927115, −4.89653276676034477975892816512, −4.49967556618138876296694104667, −4.27835681103880114877619099807, −4.24347609946144078514683546505, −4.00047488242436642584775203610, −3.80851351160591977331227038068, −3.39792750416127745301261692097, −3.18043529260470390889008086896, −2.81334936367720578112579217090, −2.68597898394885858123676013262, −2.61626270116687357620757998983, −2.21253644858914769900581490115, −2.21132718150798341914396701885, −1.73702970571412030960038402410, −1.16153817991010814552995330480, 0, 0, 0, 0,
1.16153817991010814552995330480, 1.73702970571412030960038402410, 2.21132718150798341914396701885, 2.21253644858914769900581490115, 2.61626270116687357620757998983, 2.68597898394885858123676013262, 2.81334936367720578112579217090, 3.18043529260470390889008086896, 3.39792750416127745301261692097, 3.80851351160591977331227038068, 4.00047488242436642584775203610, 4.24347609946144078514683546505, 4.27835681103880114877619099807, 4.49967556618138876296694104667, 4.89653276676034477975892816512, 5.18249385517977865122126927115, 5.21583453749440559231278847871, 5.26514233848340792891083683181, 5.73734517382288182070797573711, 5.82287036279973228969756087985, 6.01038533008779990631292437217, 6.38811075358275037106605262539, 6.44113201758756424590034046968, 6.55258345994619908649293073884, 6.96154377683027520326127900211
