| L(s) = 1 | − 4·2-s − 6·3-s + 4·4-s − 12·5-s + 24·6-s + 16·8-s + 9·9-s + 48·10-s − 4·11-s − 24·12-s + 96·13-s + 72·15-s − 64·16-s − 132·17-s − 36·18-s − 120·19-s − 48·20-s + 16·22-s + 76·23-s − 96·24-s + 284·25-s − 384·26-s + 54·27-s − 224·29-s − 288·30-s − 432·31-s + 64·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.07·5-s + 1.63·6-s + 0.707·8-s + 1/3·9-s + 1.51·10-s − 0.109·11-s − 0.577·12-s + 2.04·13-s + 1.23·15-s − 16-s − 1.88·17-s − 0.471·18-s − 1.44·19-s − 0.536·20-s + 0.155·22-s + 0.689·23-s − 0.816·24-s + 2.27·25-s − 2.89·26-s + 0.384·27-s − 1.43·29-s − 1.75·30-s − 2.50·31-s + 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.001053560801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001053560801\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T - 28 p T^{2} + 408 T^{3} + 47031 T^{4} + 408 p^{3} T^{5} - 28 p^{7} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2578 T^{2} - 272 T^{3} + 4935979 T^{4} - 272 p^{3} T^{5} - 2578 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 48 T + 4520 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 132 T + 3484 T^{2} + 31944 p T^{3} + 292671 p^{2} T^{4} + 31944 p^{4} T^{5} + 3484 p^{6} T^{6} + 132 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 120 T + 1314 T^{2} - 75840 T^{3} + 25427915 T^{4} - 75840 p^{3} T^{5} + 1314 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 76 T + 806 T^{2} + 1471664 T^{3} - 193611581 T^{4} + 1471664 p^{3} T^{5} + 806 p^{6} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 112 T + 6914 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 432 T + 86218 T^{2} + 17635968 T^{3} + 3634145571 T^{4} + 17635968 p^{3} T^{5} + 86218 p^{6} T^{6} + 432 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 280 T + 22294 T^{2} + 12656000 T^{3} - 3389038373 T^{4} + 12656000 p^{3} T^{5} + 22294 p^{6} T^{6} - 280 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 36 T + 105908 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 128 T - 2778 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 264 T + 73114 T^{2} + 55720896 T^{3} - 18003580413 T^{4} + 55720896 p^{3} T^{5} + 73114 p^{6} T^{6} - 264 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 268 T - 170158 T^{2} - 14946896 T^{3} + 25697985547 T^{4} - 14946896 p^{3} T^{5} - 170158 p^{6} T^{6} + 268 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 336 T - 148478 T^{2} - 50193024 T^{3} + 2949366651 T^{4} - 50193024 p^{3} T^{5} - 148478 p^{6} T^{6} + 336 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 504 T - 14232 T^{2} + 93599856 T^{3} - 37220190553 T^{4} + 93599856 p^{3} T^{5} - 14232 p^{6} T^{6} - 504 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 384 T - 449462 T^{2} + 1769472 T^{3} + 221503407627 T^{4} + 1769472 p^{3} T^{5} - 449462 p^{6} T^{6} - 384 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 396 T + 521098 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 312 T + 3024 T^{2} + 213318768 T^{3} - 180306434737 T^{4} + 213318768 p^{3} T^{5} + 3024 p^{6} T^{6} - 312 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 848 T - 266750 T^{2} + 189952 T^{3} + 374274336739 T^{4} + 189952 p^{3} T^{5} - 266750 p^{6} T^{6} - 848 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 648 T + 1235750 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 612 T - 1067780 T^{2} - 19820232 T^{3} + 1319275320879 T^{4} - 19820232 p^{3} T^{5} - 1067780 p^{6} T^{6} - 612 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2184 T + 2982432 T^{2} + 2184 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.238633260271698171702504594253, −7.81948518195331734908697103267, −7.77106801055823858706039026568, −7.33450809360205102366023042612, −7.08394673065323396971392043583, −6.86055737871897503984086352813, −6.52433066567771156644882290009, −6.46034210490824823409009800311, −6.14419450477495134192734546752, −5.64491628838590276655365555977, −5.56434141907230460663369170930, −5.10669363235352891889385624852, −4.97365926057605728641642837716, −4.34670945592688112463290671904, −4.20237759534950672199099351072, −4.15919475010250069959434393158, −3.56542141614852808616657614039, −3.44287607904797674857479021146, −2.76871098709291140076328244499, −2.46212399574432505752428575195, −1.72051441086490872413057017539, −1.65773691382372707941037784954, −0.988309156185134890350063563244, −0.59439002931141291112603504723, −0.01379998886448811030699679187,
0.01379998886448811030699679187, 0.59439002931141291112603504723, 0.988309156185134890350063563244, 1.65773691382372707941037784954, 1.72051441086490872413057017539, 2.46212399574432505752428575195, 2.76871098709291140076328244499, 3.44287607904797674857479021146, 3.56542141614852808616657614039, 4.15919475010250069959434393158, 4.20237759534950672199099351072, 4.34670945592688112463290671904, 4.97365926057605728641642837716, 5.10669363235352891889385624852, 5.56434141907230460663369170930, 5.64491628838590276655365555977, 6.14419450477495134192734546752, 6.46034210490824823409009800311, 6.52433066567771156644882290009, 6.86055737871897503984086352813, 7.08394673065323396971392043583, 7.33450809360205102366023042612, 7.77106801055823858706039026568, 7.81948518195331734908697103267, 8.238633260271698171702504594253
