| L(s) = 1 | + 2·2-s + 4-s − 4·5-s − 2·7-s − 2·8-s − 8·10-s + 2·11-s − 4·13-s − 4·14-s − 4·16-s − 4·17-s + 4·19-s − 4·20-s + 4·22-s − 4·23-s + 12·25-s − 8·26-s − 2·28-s + 12·29-s + 8·31-s − 2·32-s − 8·34-s + 8·35-s + 16·37-s + 8·38-s + 8·40-s + 16·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 1.78·5-s − 0.755·7-s − 0.707·8-s − 2.52·10-s + 0.603·11-s − 1.10·13-s − 1.06·14-s − 16-s − 0.970·17-s + 0.917·19-s − 0.894·20-s + 0.852·22-s − 0.834·23-s + 12/5·25-s − 1.56·26-s − 0.377·28-s + 2.22·29-s + 1.43·31-s − 0.353·32-s − 1.37·34-s + 1.35·35-s + 2.63·37-s + 1.29·38-s + 1.26·40-s + 2.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.630611779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.630611779\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T + 10 T^{2} - 112 T^{3} - 525 T^{4} - 112 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 24 T^{2} - 8 T^{3} + 935 T^{4} - 8 p T^{5} - 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 56 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 16 T + 120 T^{2} - 992 T^{3} + 7655 T^{4} - 992 p T^{5} + 120 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 272 T^{3} - 2285 T^{4} - 272 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 376 T^{3} - 3513 T^{4} - 376 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 31 T^{2} + 658 T^{3} + 12180 T^{4} + 658 p T^{5} + 31 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 129 T^{2} + 1314 T^{3} + 14540 T^{4} + 1314 p T^{5} + 129 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 434 p T^{5} + 31 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 48 T^{2} - 992 T^{3} + 19247 T^{4} - 992 p T^{5} + 48 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 103 T^{2} - 1134 T^{3} + 17004 T^{4} - 1134 p T^{5} + 103 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 448 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.81841294357863948575366180259, −6.31169845196690366901169820586, −6.30926385504824540853227487173, −6.20405512871942641918250070681, −6.13865218717778328462025436062, −5.90730961329501891166193816931, −5.38039568567185409384845540927, −5.06810655698789543555797692340, −4.91187433735736398387896429515, −4.63023384219698788269409144239, −4.58507277031259446959401272697, −4.52770823154717838700104514266, −4.09586248631227615082460066769, −4.01575462380281164272132198203, −3.61974060270454149527275859121, −3.52803436411245840570195939088, −3.04187915196937989360544371497, −3.02381880984752048594946359772, −2.59587017626661169940388430426, −2.49431033654741987803175642693, −2.35221743658253561645517991563, −1.52355265928822658220569050095, −1.13235652129282728469954378969, −0.60992088554666811325275681464, −0.49158167002211527605023199551,
0.49158167002211527605023199551, 0.60992088554666811325275681464, 1.13235652129282728469954378969, 1.52355265928822658220569050095, 2.35221743658253561645517991563, 2.49431033654741987803175642693, 2.59587017626661169940388430426, 3.02381880984752048594946359772, 3.04187915196937989360544371497, 3.52803436411245840570195939088, 3.61974060270454149527275859121, 4.01575462380281164272132198203, 4.09586248631227615082460066769, 4.52770823154717838700104514266, 4.58507277031259446959401272697, 4.63023384219698788269409144239, 4.91187433735736398387896429515, 5.06810655698789543555797692340, 5.38039568567185409384845540927, 5.90730961329501891166193816931, 6.13865218717778328462025436062, 6.20405512871942641918250070681, 6.30926385504824540853227487173, 6.31169845196690366901169820586, 6.81841294357863948575366180259
