| L(s) = 1 | − 132·9-s − 1.12e3·11-s − 2.00e3·19-s − 2.68e3·29-s + 4.49e3·31-s + 4.61e4·41-s + 5.65e4·49-s + 1.25e5·59-s + 2.82e4·61-s − 9.44e4·71-s − 1.31e5·79-s − 3.07e4·81-s + 1.10e5·89-s + 1.47e5·99-s + 2.05e5·101-s − 5.57e4·109-s + 1.76e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.73e5·169-s + ⋯ |
| L(s) = 1 | − 0.543·9-s − 2.79·11-s − 1.27·19-s − 0.591·29-s + 0.840·31-s + 4.28·41-s + 3.36·49-s + 4.68·59-s + 0.970·61-s − 2.22·71-s − 2.37·79-s − 0.520·81-s + 1.47·89-s + 1.51·99-s + 2.00·101-s − 0.449·109-s + 1.09·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.00·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1485185113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1485185113\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 44 p T^{2} + 5350 p^{2} T^{4} + 44 p^{11} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 8084 p T^{2} + 1352943558 T^{4} - 8084 p^{11} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 560 T + 381926 T^{2} + 560 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 373292 T^{2} + 167403787638 T^{4} - 373292 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 1613316 T^{2} + 4602934743878 T^{4} + 1613316 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 1000 T + 4533462 T^{2} + 1000 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 7126092 T^{2} + 48612883261958 T^{4} - 7126092 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1340 T - 8758306 T^{2} + 1340 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2248 T + 57017022 T^{2} - 2248 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 211585036 T^{2} + 19959790722550422 T^{4} - 211585036 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 313073372 T^{2} + 49182412950657078 T^{4} - 313073372 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 104863596 T^{2} + 104529727880710502 T^{4} - 104863596 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1357205900 T^{2} + 805869805574922198 T^{4} - 1357205900 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 62584 T + 2277965606 T^{2} - 62584 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 14108 T + 1110042462 T^{2} - 14108 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1448611388 T^{2} + 3060385933795078038 T^{4} - 1448611388 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 47208 T + 4011779662 T^{2} + 47208 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4251788572 T^{2} + 9098112686809466598 T^{4} - 4251788572 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 65904 T + 3994274078 T^{2} + 65904 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9324010812 T^{2} + 49682906641812448598 T^{4} - 9324010812 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 55020 T + 10818978262 T^{2} - 55020 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1419021116 T^{2} - 92174956617487206138 T^{4} - 1419021116 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.37307672484729644342613798632, −7.19793958871723590049513432859, −6.79641939462733697407627998063, −6.68781920614316502626505503896, −6.21714727104239879680102179070, −5.79422326553415760912951776346, −5.74171955167324646849867698644, −5.63030960014733504128481092149, −5.33147719771114408454969296635, −5.23454829926243579760615188934, −4.50371632290345239577125651660, −4.45389882057029468973322330317, −4.25510654705136452631931707885, −3.97520514206760639738487698235, −3.63285294921867625123270432616, −3.06889897384129707026770485874, −2.85181799238527380361650591663, −2.58056705762740563015287731042, −2.36164130550640247167346062190, −2.17472251969225840768940775764, −1.92542722407541760351906555412, −0.980261759313446434123884768672, −0.858948845768503965434699113946, −0.66491686848561790630722575290, −0.05286702441760970953360181297,
0.05286702441760970953360181297, 0.66491686848561790630722575290, 0.858948845768503965434699113946, 0.980261759313446434123884768672, 1.92542722407541760351906555412, 2.17472251969225840768940775764, 2.36164130550640247167346062190, 2.58056705762740563015287731042, 2.85181799238527380361650591663, 3.06889897384129707026770485874, 3.63285294921867625123270432616, 3.97520514206760639738487698235, 4.25510654705136452631931707885, 4.45389882057029468973322330317, 4.50371632290345239577125651660, 5.23454829926243579760615188934, 5.33147719771114408454969296635, 5.63030960014733504128481092149, 5.74171955167324646849867698644, 5.79422326553415760912951776346, 6.21714727104239879680102179070, 6.68781920614316502626505503896, 6.79641939462733697407627998063, 7.19793958871723590049513432859, 7.37307672484729644342613798632
