| L(s) = 1 | + 1.74e3·4-s + 1.69e4·9-s + 3.71e4·11-s + 1.76e6·16-s + 2.86e5·19-s + 2.31e7·29-s − 7.90e6·31-s + 2.95e7·36-s + 2.11e6·41-s + 6.47e7·44-s − 1.15e7·49-s + 2.32e8·59-s − 8.93e7·61-s + 1.29e9·64-s − 5.88e8·71-s + 4.99e8·76-s + 1.38e9·79-s + 1.46e8·81-s + 1.55e9·89-s + 6.28e8·99-s + 3.87e9·101-s − 1.22e8·109-s + 4.03e10·116-s − 7.23e9·121-s − 1.37e10·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3.40·4-s + 0.860·9-s + 0.764·11-s + 6.71·16-s + 0.504·19-s + 6.07·29-s − 1.53·31-s + 2.93·36-s + 0.117·41-s + 2.60·44-s − 2/7·49-s + 2.49·59-s − 0.826·61-s + 9.66·64-s − 2.74·71-s + 1.71·76-s + 4.00·79-s + 0.378·81-s + 2.63·89-s + 0.658·99-s + 3.70·101-s − 0.0829·109-s + 20.7·116-s − 3.06·121-s − 5.23·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(58.11160800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(58.11160800\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 109 p^{4} T^{2} + 20001 p^{6} T^{4} - 109 p^{22} T^{6} + p^{36} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 1882 p^{2} T^{2} + 1731931 p^{4} T^{4} - 1882 p^{20} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 18566 T + 4136090463 T^{2} - 18566 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 35734734546 T^{2} + \)\(53\!\cdots\!87\)\( T^{4} - 35734734546 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 400757328402 T^{2} + \)\(68\!\cdots\!19\)\( T^{4} - 400757328402 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 143276 T + 642750798250 T^{2} - 143276 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3716115300204 T^{2} + \)\(85\!\cdots\!74\)\( T^{4} + 3716115300204 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 399226 p T + 61819642147595 T^{2} - 399226 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 3953760 T + 18380996896542 T^{2} + 3953760 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 259084528777036 T^{2} + \)\(49\!\cdots\!82\)\( T^{4} - 259084528777036 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 1058992 T - 202350146478094 T^{2} - 1058992 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 369777224136228 T^{2} + \)\(34\!\cdots\!94\)\( T^{4} - 369777224136228 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 893470329422474 T^{2} - \)\(37\!\cdots\!53\)\( T^{4} - 893470329422474 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12441509189691476 T^{2} + \)\(60\!\cdots\!74\)\( T^{4} - 12441509189691476 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 116159208 T + 8473255943386694 T^{2} - 116159208 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 44688544 T + 21891928402164378 T^{2} + 44688544 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9140919712965644 T^{2} + \)\(85\!\cdots\!14\)\( T^{4} - 9140919712965644 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 294165824 T + 66419314231297806 T^{2} + 294165824 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 228910758186064924 T^{2} + \)\(20\!\cdots\!90\)\( T^{4} - 228910758186064924 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 692852854 T + 353229306838520119 T^{2} - 692852854 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 465453587192559020 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - 465453587192559020 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 779043704 T + 776620283810146850 T^{2} - 779043704 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 565034345617929950 T^{2} + \)\(11\!\cdots\!03\)\( T^{4} + 565034345617929950 p^{18} T^{6} + p^{36} T^{8} \) |
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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
−7.60285418988175606957842483149, −6.92119247366971812386481974039, −6.91762203136181381388601313142, −6.91083334739397695704854408381, −6.62620416670885047730093130295, −6.33794564128637430492093249641, −6.05574910262246413674219192346, −5.93508639461919858415672738119, −5.36426541457156286212461751770, −5.20537749003550877479625152448, −4.69411050701293414128489150857, −4.34092311331886636031295134738, −4.33104031023771865912467563076, −3.57334522278522012608041365580, −3.20895386364833060935505400399, −3.09161937432092919211061403237, −3.05331539304796349699928701768, −2.41256638910707229372572248515, −2.16403573641047107355625806084, −1.93216098595404481962751743818, −1.81527816743184604972810723403, −1.09698891795236830510446499656, −1.03319722145540612297934964676, −0.850483780253857071118536812771, −0.53416562295898580710913470943,
0.53416562295898580710913470943, 0.850483780253857071118536812771, 1.03319722145540612297934964676, 1.09698891795236830510446499656, 1.81527816743184604972810723403, 1.93216098595404481962751743818, 2.16403573641047107355625806084, 2.41256638910707229372572248515, 3.05331539304796349699928701768, 3.09161937432092919211061403237, 3.20895386364833060935505400399, 3.57334522278522012608041365580, 4.33104031023771865912467563076, 4.34092311331886636031295134738, 4.69411050701293414128489150857, 5.20537749003550877479625152448, 5.36426541457156286212461751770, 5.93508639461919858415672738119, 6.05574910262246413674219192346, 6.33794564128637430492093249641, 6.62620416670885047730093130295, 6.91083334739397695704854408381, 6.91762203136181381388601313142, 6.92119247366971812386481974039, 7.60285418988175606957842483149
