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
\(L(1)\) |
\(\approx\) |
\(7.421708168\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.421708168\) |
\(L(\frac{3}{2})\) |
|
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
−8.145937690190340947655480001157, −7.83830983452160466298215998408, −7.82352221826085814684061084194, −7.59181757350155572662655794130, −7.47950501562766099494543993519, −7.11357815209502445875749596586, −6.94981517359820893121793937667, −6.79455537371617135669563497986, −6.38633885221180702233745407687, −5.65495942503521892204232608008, −5.58898548280695186625025921196, −5.57865861726097544518528573132, −5.37346300531111897966255538084, −4.92389216227506731130061848997, −4.50088945490560284169222347052, −4.27874832621618955336960424432, −3.85104856109090442555194094769, −3.75399559841382008852367090320, −3.56189248689979180419366785891, −3.28817301954301117711180868348, −3.27743233855085841537162551341, −2.78252869691569325060875633182, −2.14993086978748661034891363599, −1.30630764907951352205816814227, −0.863144402808129779229186426302,
0.863144402808129779229186426302, 1.30630764907951352205816814227, 2.14993086978748661034891363599, 2.78252869691569325060875633182, 3.27743233855085841537162551341, 3.28817301954301117711180868348, 3.56189248689979180419366785891, 3.75399559841382008852367090320, 3.85104856109090442555194094769, 4.27874832621618955336960424432, 4.50088945490560284169222347052, 4.92389216227506731130061848997, 5.37346300531111897966255538084, 5.57865861726097544518528573132, 5.58898548280695186625025921196, 5.65495942503521892204232608008, 6.38633885221180702233745407687, 6.79455537371617135669563497986, 6.94981517359820893121793937667, 7.11357815209502445875749596586, 7.47950501562766099494543993519, 7.59181757350155572662655794130, 7.82352221826085814684061084194, 7.83830983452160466298215998408, 8.145937690190340947655480001157