| L(s) = 1 | − 1.06e3·5-s + 9.25e4·13-s − 2.48e5·17-s + 5.45e5·25-s + 4.90e5·29-s − 3.18e6·37-s + 6.96e6·41-s + 5.59e6·49-s + 2.32e6·53-s + 3.89e7·61-s − 9.85e7·65-s + 2.97e7·73-s + 2.64e8·85-s + 4.72e7·89-s − 3.95e8·97-s + 1.98e8·101-s + 3.26e7·109-s + 2.86e8·113-s + 6.55e8·121-s − 5.34e8·125-s + ⋯ |
| L(s) = 1 | − 1.70·5-s + 3.24·13-s − 2.97·17-s + 1.39·25-s + 0.693·29-s − 1.69·37-s + 2.46·41-s + 0.969·49-s + 0.294·53-s + 2.81·61-s − 5.51·65-s + 1.04·73-s + 5.06·85-s + 0.753·89-s − 4.47·97-s + 1.90·101-s + 0.231·109-s + 1.75·113-s + 3.05·121-s − 2.18·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3840556830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3840556830\) |
| \(L(5)\) |
|
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 | $D_{4}$ | \( ( 1 + 532 T + 30318 p T^{2} + 532 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 114116 p^{2} T^{2} + 28741685190 p^{4} T^{4} - 114116 p^{18} T^{6} + p^{32} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 655420004 T^{2} + 189840211843101510 T^{4} - 655420004 p^{16} T^{6} + p^{32} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 46292 T + 2048828454 T^{2} - 46292 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 124260 T + 12631375046 T^{2} + 124260 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2327266444 p T^{2} + 2620064808046455126 p^{2} T^{4} - 2327266444 p^{17} T^{6} + p^{32} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 22564164740 T^{2} + \)\(26\!\cdots\!58\)\( T^{4} - 22564164740 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 245356 T + 187952884902 T^{2} - 245356 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2308308775940 T^{2} + \)\(28\!\cdots\!38\)\( p^{2} T^{4} - 2308308775940 p^{16} T^{6} + p^{32} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1590444 T + 4350022314662 T^{2} + 1590444 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 3482044 T + 18665604055302 T^{2} - 3482044 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 17226221238172 T^{2} + \)\(34\!\cdots\!54\)\( T^{4} + 17226221238172 p^{16} T^{6} + p^{32} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 19635696904196 T^{2} + \)\(12\!\cdots\!46\)\( T^{4} - 19635696904196 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 1163052 T + 118473709037222 T^{2} - 1163052 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 462492700881764 T^{2} + \)\(93\!\cdots\!30\)\( T^{4} - 462492700881764 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 19491284 T + 465046980345126 T^{2} - 19491284 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 523122408286876 T^{2} + \)\(14\!\cdots\!10\)\( T^{4} + 523122408286876 p^{16} T^{6} + p^{32} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 2546570207785604 T^{2} + \)\(24\!\cdots\!10\)\( T^{4} - 2546570207785604 p^{16} T^{6} + p^{32} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 14888676 T + 1528099309486406 T^{2} - 14888676 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 3114613788648452 T^{2} + \)\(67\!\cdots\!18\)\( T^{4} - 3114613788648452 p^{16} T^{6} + p^{32} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4124771092248164 T^{2} + \)\(84\!\cdots\!86\)\( T^{4} - 4124771092248164 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 23623452 T + 7124716017813062 T^{2} - 23623452 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 197989660 T + 24238250475556038 T^{2} + 197989660 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
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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
−7.01187142443314806892596945566, −6.91722375697760622239782320912, −6.86479883792267060693224403617, −6.35319587921509076804692213062, −6.02217652576559421933896740414, −5.92173237066663020526958572708, −5.83127842269587337610259253322, −5.27687396098015155152152721659, −4.73824785341139214173580810589, −4.67198145277131244818272682279, −4.60386529630814837678230750149, −3.90529598847943296657328366775, −3.85955075851228163989323769937, −3.69142241037131725145530410146, −3.68618868126734488137212694306, −3.17465047458119340745868199525, −2.55824325039877168968052060738, −2.43093251916157878542276380730, −2.27047817068172844537582972562, −1.78207604968707652458362514922, −1.24571438051978166312067987221, −1.03888847405398797813230087530, −0.951930472404065823090105374900, −0.44643926183426765360762972470, −0.083082797276033757805482074340,
0.083082797276033757805482074340, 0.44643926183426765360762972470, 0.951930472404065823090105374900, 1.03888847405398797813230087530, 1.24571438051978166312067987221, 1.78207604968707652458362514922, 2.27047817068172844537582972562, 2.43093251916157878542276380730, 2.55824325039877168968052060738, 3.17465047458119340745868199525, 3.68618868126734488137212694306, 3.69142241037131725145530410146, 3.85955075851228163989323769937, 3.90529598847943296657328366775, 4.60386529630814837678230750149, 4.67198145277131244818272682279, 4.73824785341139214173580810589, 5.27687396098015155152152721659, 5.83127842269587337610259253322, 5.92173237066663020526958572708, 6.02217652576559421933896740414, 6.35319587921509076804692213062, 6.86479883792267060693224403617, 6.91722375697760622239782320912, 7.01187142443314806892596945566
