| L(s) = 1 | − 32·5-s − 296·13-s + 576·17-s − 420·25-s + 3.48e3·29-s + 1.32e3·37-s + 6.84e3·41-s + 3.04e3·49-s + 1.10e4·53-s + 424·61-s + 9.47e3·65-s + 3.00e3·73-s − 1.84e4·85-s − 384·89-s + 1.11e4·97-s − 2.91e3·101-s + 1.86e4·109-s + 2.17e3·113-s + 2.73e4·121-s + 2.06e4·125-s + ⋯ |
| L(s) = 1 | − 1.27·5-s − 1.75·13-s + 1.99·17-s − 0.671·25-s + 4.14·29-s + 0.964·37-s + 4.07·41-s + 1.26·49-s + 3.93·53-s + 0.113·61-s + 2.24·65-s + 0.562·73-s − 2.55·85-s − 0.0484·89-s + 1.18·97-s − 0.285·101-s + 1.56·109-s + 0.170·113-s + 1.86·121-s + 1.32·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.623404689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.623404689\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 16 T + 594 T^{2} + 16 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3044 T^{2} + 9238086 T^{4} - 3044 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 27332 T^{2} + 426733638 T^{4} - 27332 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 148 T + 16518 T^{2} + 148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 288 T + 184898 T^{2} - 288 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 497092 T^{2} + 265140438 p^{2} T^{4} - 497092 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 257156 T^{2} + 145975182726 T^{4} - 257156 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1744 T + 2053266 T^{2} - 1744 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2943332 T^{2} + 3834092018118 T^{4} - 2943332 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 660 T + 2705222 T^{2} - 660 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 3424 T + 8095746 T^{2} - 3424 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6217796 T^{2} + 20023638876486 T^{4} - 6217796 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3265276 T^{2} + 3102160162566 T^{4} + 3265276 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 5520 T + 19769042 T^{2} - 5520 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4500932 T^{2} - 162959688247482 T^{4} - 4500932 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 212 T + 27288198 T^{2} - 212 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 15078212 T^{2} + 51214632004038 T^{4} - 15078212 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 24936836 T^{2} + 605069020552326 T^{4} - 24936836 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 1500 T + 57174662 T^{2} - 1500 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 45515876 T^{2} + 513196620838086 T^{4} - 45515876 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 134137412 T^{2} + 8408135231992518 T^{4} - 134137412 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 192 T + 30096578 T^{2} + 192 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 5564 T + 18910086 T^{2} - 5564 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−7.991167914924184180547721213778, −7.47578972644485694164993028493, −7.42969668477659085000243508977, −7.36537120629971482292093160155, −7.00796997288028794946707956940, −6.78260945635726946282796911538, −6.15223812073666247854511161374, −5.97881601399155091228154349704, −5.92143057774019888002533238730, −5.60863593544882314276556436344, −5.00345718495702324384080358894, −4.93949868232261897672738727192, −4.58880018203434786875419547981, −4.38882828583970856082423825534, −3.95077045362829066328973052715, −3.73201346524809136931817352196, −3.57171653056850152002051487610, −2.82933724005438392660666299738, −2.59269387163415603676605122657, −2.54489176838542221683344611287, −2.22602259003357516068134857289, −1.26348854679491824277439890187, −0.850732300979571746431442935904, −0.811983526796628426162440536347, −0.41455717528071116592621064081,
0.41455717528071116592621064081, 0.811983526796628426162440536347, 0.850732300979571746431442935904, 1.26348854679491824277439890187, 2.22602259003357516068134857289, 2.54489176838542221683344611287, 2.59269387163415603676605122657, 2.82933724005438392660666299738, 3.57171653056850152002051487610, 3.73201346524809136931817352196, 3.95077045362829066328973052715, 4.38882828583970856082423825534, 4.58880018203434786875419547981, 4.93949868232261897672738727192, 5.00345718495702324384080358894, 5.60863593544882314276556436344, 5.92143057774019888002533238730, 5.97881601399155091228154349704, 6.15223812073666247854511161374, 6.78260945635726946282796911538, 7.00796997288028794946707956940, 7.36537120629971482292093160155, 7.42969668477659085000243508977, 7.47578972644485694164993028493, 7.991167914924184180547721213778
