| L(s) = 1 | + 8·5-s − 2·9-s − 12·17-s + 20·25-s + 12·29-s + 24·37-s + 8·41-s − 16·45-s − 6·49-s + 24·53-s + 12·61-s + 24·73-s − 3·81-s − 96·85-s + 16·89-s + 48·97-s + 36·101-s − 24·109-s − 36·113-s − 14·121-s − 40·125-s + 127-s + 131-s + 137-s + 139-s + 96·145-s + 149-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 2/3·9-s − 2.91·17-s + 4·25-s + 2.22·29-s + 3.94·37-s + 1.24·41-s − 2.38·45-s − 6/7·49-s + 3.29·53-s + 1.53·61-s + 2.80·73-s − 1/3·81-s − 10.4·85-s + 1.69·89-s + 4.87·97-s + 3.58·101-s − 2.29·109-s − 3.38·113-s − 1.27·121-s − 3.57·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.97·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.69929075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.69929075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 + 2 T^{2} + 7 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 6 T^{2} - T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 14 T^{2} + 183 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 54 T^{2} + 1343 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 10 T^{2} + 975 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 36 T^{2} + 518 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 82 T^{2} + 4407 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 4 T^{2} + 2694 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 10 T^{2} + 6015 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 246 T^{2} + 23999 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 14 T^{2} + 1383 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 196 T^{2} + 20358 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 212 T^{2} + 23286 T^{4} + 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 8 T + 167 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 24 T + 335 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.94184664639347053353034240805, −5.61577237962366390032595437572, −5.33956919858318088365796375992, −5.28108712777309370804236854173, −5.12677623494920046051003888956, −4.79775757919956575384287121926, −4.75654032116821498723860616807, −4.37038169905296304757194008329, −4.26664682453302570013059360108, −4.04945311624170794286067242408, −3.90142523435735189921468561418, −3.58056082701344505131959704587, −3.41393134280874500600501082507, −2.77429132083988725189528469093, −2.74533127673538015364014662278, −2.55163948399715557602002178049, −2.48868376445881194554485907161, −2.27227925235580273776365267121, −2.05611678814104365367289862196, −1.85615978087165839844504469436, −1.80155011021054350362900125685, −1.21248115921947163502407719856, −0.893703462753866922575439030569, −0.70393521010110959464768181263, −0.46826868496538690293396876483,
0.46826868496538690293396876483, 0.70393521010110959464768181263, 0.893703462753866922575439030569, 1.21248115921947163502407719856, 1.80155011021054350362900125685, 1.85615978087165839844504469436, 2.05611678814104365367289862196, 2.27227925235580273776365267121, 2.48868376445881194554485907161, 2.55163948399715557602002178049, 2.74533127673538015364014662278, 2.77429132083988725189528469093, 3.41393134280874500600501082507, 3.58056082701344505131959704587, 3.90142523435735189921468561418, 4.04945311624170794286067242408, 4.26664682453302570013059360108, 4.37038169905296304757194008329, 4.75654032116821498723860616807, 4.79775757919956575384287121926, 5.12677623494920046051003888956, 5.28108712777309370804236854173, 5.33956919858318088365796375992, 5.61577237962366390032595437572, 5.94184664639347053353034240805
