| L(s) = 1 | − 2·4-s − 4·5-s + 3·16-s + 8·20-s + 2·25-s + 12·29-s − 16·31-s + 16·41-s + 2·49-s + 24·59-s − 4·64-s − 24·71-s + 32·79-s − 12·80-s + 24·89-s − 4·100-s + 16·101-s − 16·109-s − 24·116-s − 38·121-s + 32·124-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s + 3/4·16-s + 1.78·20-s + 2/5·25-s + 2.22·29-s − 2.87·31-s + 2.49·41-s + 2/7·49-s + 3.12·59-s − 1/2·64-s − 2.84·71-s + 3.60·79-s − 1.34·80-s + 2.54·89-s − 2/5·100-s + 1.59·101-s − 1.53·109-s − 2.22·116-s − 3.45·121-s + 2.87·124-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9487608053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9487608053\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T^{2} + 51 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 315 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 71 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 6291 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1670 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 198 T^{2} + 15371 T^{4} - 198 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 22710 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4230 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 106 T^{2} + 17739 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.02573289138485796378101316006, −5.91079321144527856732438140446, −5.60528249995130852028694832740, −5.44010686247472509460860168513, −5.27900172217678510638033731009, −5.17188026032314241446046205277, −4.83855690282284585175866506531, −4.49518449103386073933436799969, −4.39716141530395578948205897324, −4.31986622155423237050830031500, −4.13695351712228260690119464596, −3.79173530499187407756904169797, −3.59668021517392531234737875803, −3.46859734787065875664228417706, −3.44665292731848628477045373684, −3.00489844789573925640939861636, −2.70331145930435616757934117590, −2.36413637532014137399237414288, −2.27186861230986432627170657236, −2.00082558545639453574563306795, −1.55433838949197417017136989419, −1.07907748249255115341260539784, −0.948968486030020431898171832621, −0.55125228426889112981815686767, −0.24055065793603950669518480481,
0.24055065793603950669518480481, 0.55125228426889112981815686767, 0.948968486030020431898171832621, 1.07907748249255115341260539784, 1.55433838949197417017136989419, 2.00082558545639453574563306795, 2.27186861230986432627170657236, 2.36413637532014137399237414288, 2.70331145930435616757934117590, 3.00489844789573925640939861636, 3.44665292731848628477045373684, 3.46859734787065875664228417706, 3.59668021517392531234737875803, 3.79173530499187407756904169797, 4.13695351712228260690119464596, 4.31986622155423237050830031500, 4.39716141530395578948205897324, 4.49518449103386073933436799969, 4.83855690282284585175866506531, 5.17188026032314241446046205277, 5.27900172217678510638033731009, 5.44010686247472509460860168513, 5.60528249995130852028694832740, 5.91079321144527856732438140446, 6.02573289138485796378101316006
