| L(s) = 1 | + 4·4-s + 4·16-s + 16·19-s + 16·41-s + 64·59-s + 24·61-s − 16·64-s + 64·76-s + 32·79-s − 16·101-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 64·164-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·4-s + 16-s + 3.67·19-s + 2.49·41-s + 8.33·59-s + 3.07·61-s − 2·64-s + 7.34·76-s + 3.60·79-s − 1.59·101-s + 6.54·121-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 + 4.99·164-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(27.60243657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.60243657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 p^{2} T^{4} + 7011 T^{8} - 2 p^{6} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 194 T^{4} + 171 p^{2} T^{8} - 194 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 68 T^{4} - 434490 T^{8} + 68 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 60 T^{2} + 1814 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 110 T^{2} + 4899 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 868 T^{4} + 2707878 T^{8} + 868 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 3358 T^{4} + 8633475 T^{8} + 3358 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 5188 T^{4} + 13723398 T^{8} + 5188 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 + 5348 T^{4} + 16710438 T^{8} + 5348 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 16 T + 170 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 67 | \( 1 - 6818 T^{4} + 25780611 T^{8} - 6818 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 188 T^{2} + 17190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 6242 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 - 284 T^{4} - 90968346 T^{8} - 284 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 3842 T^{4} + 131200515 T^{8} - 3842 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.25378462640446652994271272273, −4.18411493053038748698204779333, −4.13358358507043527774673227393, −4.01854176721297901638358410990, −3.85017264616024010970886837377, −3.48847857371927639188247029619, −3.34573903607270297299181285717, −3.29878623752087529210903791581, −3.27383995795580997572537787831, −3.24369596551973188111339111936, −3.18968454398108628724799937484, −2.70462178886949743011580928214, −2.48823898113801802743683226374, −2.41921483946898846923152881697, −2.32393355316460838986935809704, −2.29156355337938149730520093586, −2.22877566701772899960449544897, −1.87899933841634962951419531677, −1.78890961001234894095448994371, −1.62827877062494839081592894571, −1.07084534081007234801258488321, −0.950347979051671614902603423296, −0.921786960479907610881697494765, −0.68068694918993886199676050362, −0.62405684757798701195565561912,
0.62405684757798701195565561912, 0.68068694918993886199676050362, 0.921786960479907610881697494765, 0.950347979051671614902603423296, 1.07084534081007234801258488321, 1.62827877062494839081592894571, 1.78890961001234894095448994371, 1.87899933841634962951419531677, 2.22877566701772899960449544897, 2.29156355337938149730520093586, 2.32393355316460838986935809704, 2.41921483946898846923152881697, 2.48823898113801802743683226374, 2.70462178886949743011580928214, 3.18968454398108628724799937484, 3.24369596551973188111339111936, 3.27383995795580997572537787831, 3.29878623752087529210903791581, 3.34573903607270297299181285717, 3.48847857371927639188247029619, 3.85017264616024010970886837377, 4.01854176721297901638358410990, 4.13358358507043527774673227393, 4.18411493053038748698204779333, 4.25378462640446652994271272273
Plot not available for L-functions of degree greater than 10.
