| L(s) = 1 | + 4·2-s + 10·4-s + 8·5-s + 2·7-s + 20·8-s + 2·9-s + 32·10-s + 8·14-s + 35·16-s − 12·17-s + 8·18-s − 16·19-s + 80·20-s + 2·23-s + 20·25-s + 20·28-s − 8·29-s + 56·32-s − 48·34-s + 16·35-s + 20·36-s − 64·38-s + 160·40-s + 16·45-s + 8·46-s + 6·49-s + 80·50-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 3.57·5-s + 0.755·7-s + 7.07·8-s + 2/3·9-s + 10.1·10-s + 2.13·14-s + 35/4·16-s − 2.91·17-s + 1.88·18-s − 3.67·19-s + 17.8·20-s + 0.417·23-s + 4·25-s + 3.77·28-s − 1.48·29-s + 9.89·32-s − 8.23·34-s + 2.70·35-s + 10/3·36-s − 10.3·38-s + 25.2·40-s + 2.38·45-s + 1.17·46-s + 6/7·49-s + 11.3·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(24.18349085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.18349085\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T^{2} + 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 38 T^{2} + 682 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 24 T^{2} + 1998 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 134 T^{2} + 7210 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 108 T^{2} + 6006 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 162 T^{2} + 10242 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 96 T^{2} + 6654 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 22 T^{2} - 3254 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 146 T^{2} + 10914 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 210 T^{2} + 19170 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 10662 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 136 T^{2} + 11598 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.406414333879055621476288266889, −8.333948529591292030549002830157, −7.58950117178182618473318380613, −7.53356166302327413622906675204, −7.32572411231876420151329836142, −6.60720099829769521897657829383, −6.56507710054688070249345860811, −6.55815224023800587509397866822, −6.41773232087353090155441003501, −5.84190725112624711537724536002, −5.73001043340670104287773008484, −5.71140972642038346747114610085, −5.48320614247508844034154108789, −4.76301756899638923550747388137, −4.60545232195537386794198827900, −4.50037387115595650242597798419, −4.41453401120188718683933908164, −3.83642081542673910127861349121, −3.63412892568660618926005559251, −2.98937659194250358954461307853, −2.43512090514490396848984430702, −2.22456787595562881073116324856, −1.90555966774446861277939732386, −1.86656986953554832765310428626, −1.83492611549238908442854593056,
1.83492611549238908442854593056, 1.86656986953554832765310428626, 1.90555966774446861277939732386, 2.22456787595562881073116324856, 2.43512090514490396848984430702, 2.98937659194250358954461307853, 3.63412892568660618926005559251, 3.83642081542673910127861349121, 4.41453401120188718683933908164, 4.50037387115595650242597798419, 4.60545232195537386794198827900, 4.76301756899638923550747388137, 5.48320614247508844034154108789, 5.71140972642038346747114610085, 5.73001043340670104287773008484, 5.84190725112624711537724536002, 6.41773232087353090155441003501, 6.55815224023800587509397866822, 6.56507710054688070249345860811, 6.60720099829769521897657829383, 7.32572411231876420151329836142, 7.53356166302327413622906675204, 7.58950117178182618473318380613, 8.333948529591292030549002830157, 8.406414333879055621476288266889
