| L(s) = 1 | + 8·7-s + 16·13-s + 48·31-s + 16·43-s + 32·49-s + 48·61-s + 48·67-s + 40·73-s + 128·91-s − 8·97-s + 8·103-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 4.43·13-s + 8.62·31-s + 2.43·43-s + 32/7·49-s + 6.14·61-s + 5.86·67-s + 4.68·73-s + 13.4·91-s − 0.812·97-s + 0.788·103-s + 3.63·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(99.51466475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(99.51466475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 4 T + 8 T^{2} + 4 T^{3} - 62 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 20 T^{2} + 262 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 8 T + 32 T^{2} - 88 T^{3} + 238 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 316 T^{4} + 64006 T^{8} - 316 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 28 T^{2} + 598 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 158 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 80 T^{2} + 2962 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 1358 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 8 T + 32 T^{2} - 88 T^{3} - 782 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 1604 T^{4} + 4130566 T^{8} + 1604 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 180 T^{2} + 14342 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 24 T + 288 T^{2} - 2376 T^{3} + 18578 T^{4} - 2376 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 20 T + 200 T^{2} - 1660 T^{3} + 13678 T^{4} - 1660 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 76 T^{2} + 5926 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 2404 T^{4} - 43030554 T^{8} + 2404 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 176 T^{2} + 15586 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.99970934117968764846979101785, −3.68736086615940889379110098585, −3.61403818202462773374309798522, −3.55161907898791844307188370511, −3.39015230605912291563701637769, −3.34610055386329233919660175747, −3.30789658234071730428296519011, −3.17694091695072693263249888138, −2.91733790143508744891107791241, −2.48405070535088705895734804393, −2.44759269272755651224474711070, −2.39131648235953259899952343647, −2.32457849100216075874705903593, −2.24426355634344950282401776902, −2.22803375980555926138596672700, −2.01856484398229542056772841810, −1.83439643981556491452387915965, −1.34135362452321793471181020997, −1.27433448070466109689304502248, −1.10637291333502931920810480645, −0.979819397794644166568903209832, −0.928181284542880839240623007285, −0.909345091204730979984570230632, −0.798196750920458832857328777688, −0.60031373427934393251865857819,
0.60031373427934393251865857819, 0.798196750920458832857328777688, 0.909345091204730979984570230632, 0.928181284542880839240623007285, 0.979819397794644166568903209832, 1.10637291333502931920810480645, 1.27433448070466109689304502248, 1.34135362452321793471181020997, 1.83439643981556491452387915965, 2.01856484398229542056772841810, 2.22803375980555926138596672700, 2.24426355634344950282401776902, 2.32457849100216075874705903593, 2.39131648235953259899952343647, 2.44759269272755651224474711070, 2.48405070535088705895734804393, 2.91733790143508744891107791241, 3.17694091695072693263249888138, 3.30789658234071730428296519011, 3.34610055386329233919660175747, 3.39015230605912291563701637769, 3.55161907898791844307188370511, 3.61403818202462773374309798522, 3.68736086615940889379110098585, 3.99970934117968764846979101785
Plot not available for L-functions of degree greater than 10.
