| L(s) = 1 | + 10·9-s − 40·13-s − 12·25-s − 40·37-s − 140·49-s − 360·61-s − 120·73-s + 19·81-s + 40·97-s + 280·109-s − 400·117-s + 84·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 324·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 10/9·9-s − 3.07·13-s − 0.479·25-s − 1.08·37-s − 2.85·49-s − 5.90·61-s − 1.64·73-s + 0.234·81-s + 0.412·97-s + 2.56·109-s − 3.41·117-s + 0.694·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.91·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1456398413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1456398413\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 10 p T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 282 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1222 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3138 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 310 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4290 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4218 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2838 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8950 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9282 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 11782 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13130 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10242 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
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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
−9.030268822524644012880979047626, −8.783553357866111495385807092185, −8.410912422517281773935282916688, −8.009542888860805966129963709488, −7.51621170320182060498473890205, −7.49458689386566053176912035398, −7.48087529083086005919033918143, −7.35768218992449691366728637217, −6.51538029548541644174973433940, −6.47399744890267992836552336299, −6.42965121271951234050653037521, −5.81969621941266909414881572127, −5.40499065924552829693748214946, −5.17294950887508157696896726092, −4.72283340238357699330839656954, −4.64601145253590350367853365568, −4.40894961992028510431517457752, −4.03256220575157467098998901306, −3.26774967706568332129687921590, −3.09256723396046892192849307120, −2.86409261275417217904979543644, −2.02920292395017213464743587191, −1.88794707335394461039887543644, −1.36249060372575857978381228555, −0.11136600134054828665966240242,
0.11136600134054828665966240242, 1.36249060372575857978381228555, 1.88794707335394461039887543644, 2.02920292395017213464743587191, 2.86409261275417217904979543644, 3.09256723396046892192849307120, 3.26774967706568332129687921590, 4.03256220575157467098998901306, 4.40894961992028510431517457752, 4.64601145253590350367853365568, 4.72283340238357699330839656954, 5.17294950887508157696896726092, 5.40499065924552829693748214946, 5.81969621941266909414881572127, 6.42965121271951234050653037521, 6.47399744890267992836552336299, 6.51538029548541644174973433940, 7.35768218992449691366728637217, 7.48087529083086005919033918143, 7.49458689386566053176912035398, 7.51621170320182060498473890205, 8.009542888860805966129963709488, 8.410912422517281773935282916688, 8.783553357866111495385807092185, 9.030268822524644012880979047626
