| L(s) = 1 | − 6·4-s − 16·11-s + 20·16-s − 72·17-s + 8·19-s + 16·25-s + 40·41-s − 80·43-s + 96·44-s − 14·49-s − 184·59-s − 24·64-s − 224·67-s + 432·68-s + 232·73-s − 48·76-s − 88·83-s − 312·89-s − 136·97-s − 96·100-s + 96·107-s + 304·113-s − 180·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.45·11-s + 5/4·16-s − 4.23·17-s + 8/19·19-s + 0.639·25-s + 0.975·41-s − 1.86·43-s + 2.18·44-s − 2/7·49-s − 3.11·59-s − 3/8·64-s − 3.34·67-s + 6.35·68-s + 3.17·73-s − 0.631·76-s − 1.06·83-s − 3.50·89-s − 1.40·97-s − 0.959·100-s + 0.897·107-s + 2.69·113-s − 1.48·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4342694324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4342694324\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 3 p T^{2} + p^{4} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 254 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 186 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 592 T^{2} + 143170 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 36 T + 870 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 4 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 268 T^{2} + 477286 T^{4} - 268 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1178 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1380 T^{2} + 1419974 T^{4} - 1380 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 1780 T^{2} + 2031622 T^{4} - 1780 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 20 T + 3174 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 40 T + 4090 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7492 T^{2} + 23390470 T^{4} - 7492 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1604 T^{2} - 6155034 T^{4} - 1604 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 92 T + 8836 T^{2} + 92 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 13232 T^{2} + 71110338 T^{4} - 13232 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 112 T + 11602 T^{2} + 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 9412 T^{2} + 47279686 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 116 T + 13894 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16900 T^{2} + 142880134 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 44 T + 13924 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 156 T + 20774 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 68 T + 3046 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.77112536912234569533824366697, −7.32697364304535203234088175289, −7.27653947562470879210188108019, −6.94707120769901225549838414961, −6.72370846111038681277716964338, −6.41367046605794350634538784193, −6.18995866995283941778300219265, −5.95312702469145557087826519446, −5.71555586124520424473435209145, −5.26118311518808628633465218659, −4.91864477720014140632659028792, −4.81640815089583845139797447632, −4.79711789505409460074473462068, −4.38899446260706918651556428068, −4.13807116486879062348051106538, −3.94575470737765481742716761213, −3.52649028332714898955434885055, −3.01701105617034633940617561011, −2.75792904433651014084865984477, −2.63590115193779445364392690569, −2.15559199385678359314322573754, −1.61207137594965906829614906486, −1.45385862046672583426022986894, −0.33144843050041846784517486301, −0.32296510087701871169410541065,
0.32296510087701871169410541065, 0.33144843050041846784517486301, 1.45385862046672583426022986894, 1.61207137594965906829614906486, 2.15559199385678359314322573754, 2.63590115193779445364392690569, 2.75792904433651014084865984477, 3.01701105617034633940617561011, 3.52649028332714898955434885055, 3.94575470737765481742716761213, 4.13807116486879062348051106538, 4.38899446260706918651556428068, 4.79711789505409460074473462068, 4.81640815089583845139797447632, 4.91864477720014140632659028792, 5.26118311518808628633465218659, 5.71555586124520424473435209145, 5.95312702469145557087826519446, 6.18995866995283941778300219265, 6.41367046605794350634538784193, 6.72370846111038681277716964338, 6.94707120769901225549838414961, 7.27653947562470879210188108019, 7.32697364304535203234088175289, 7.77112536912234569533824366697
