| L(s) = 1 | − 2·9-s + 20·17-s − 2·25-s − 40·41-s − 42·49-s − 8·73-s − 13·81-s − 40·89-s + 80·97-s + 40·113-s + 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 40·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 4.85·17-s − 2/5·25-s − 6.24·41-s − 6·49-s − 0.936·73-s − 1.44·81-s − 4.23·89-s + 8.12·97-s + 3.76·113-s + 5.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 3.23·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670312821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670312821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 + T^{2} )^{4} \) |
| good | 3 | \( ( 1 + T^{2} + 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + T^{2} + 8 p T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 3 p T^{2} + 198 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 30 T^{2} + 426 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 50 T^{2} + 1306 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 40 T^{2} + 1294 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 66 T^{2} + 2970 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 103 T^{2} + 4888 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 89 T^{2} + 4848 T^{4} + 89 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 21 T^{2} + 4518 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 80 T^{2} + 3118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 130 T^{2} + 9178 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + 66 T^{2} + 10026 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 237 T^{2} + 23622 T^{4} + 237 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 128 T^{2} + 8542 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 238 T^{2} + 25930 T^{4} - 238 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 10 T + 162 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 10 T + p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.87750793966268910676146041339, −3.75584890588958472980431483308, −3.67612740653664554875890283901, −3.43945031824024497061915527524, −3.42828788851242498133891765492, −3.26837144136308995711313255316, −3.26145262101698686414896295358, −3.09747188748134519879755855462, −3.07540018904822193943022771923, −2.96020137619498400375249156727, −2.70774505861967070878051740109, −2.67461910804919592598088679456, −2.33901315439299925869188263782, −2.29807874530651604608645176476, −1.88987546891993761302069116686, −1.77534981489918706258178510236, −1.65468930621483847090460952823, −1.63924122341612477122132050256, −1.57850004246862123975302697670, −1.37720040786071666885360295982, −1.26191030406533539604619599637, −0.802001423916207920060838061641, −0.60595501940756060486843637910, −0.50880495082752674986647491516, −0.13065704017761341941038768775,
0.13065704017761341941038768775, 0.50880495082752674986647491516, 0.60595501940756060486843637910, 0.802001423916207920060838061641, 1.26191030406533539604619599637, 1.37720040786071666885360295982, 1.57850004246862123975302697670, 1.63924122341612477122132050256, 1.65468930621483847090460952823, 1.77534981489918706258178510236, 1.88987546891993761302069116686, 2.29807874530651604608645176476, 2.33901315439299925869188263782, 2.67461910804919592598088679456, 2.70774505861967070878051740109, 2.96020137619498400375249156727, 3.07540018904822193943022771923, 3.09747188748134519879755855462, 3.26145262101698686414896295358, 3.26837144136308995711313255316, 3.42828788851242498133891765492, 3.43945031824024497061915527524, 3.67612740653664554875890283901, 3.75584890588958472980431483308, 3.87750793966268910676146041339
Plot not available for L-functions of degree greater than 10.
