| L(s) = 1 | + 4·5-s + 6·9-s + 20·11-s + 52·23-s − 84·25-s − 16·31-s − 64·37-s + 24·45-s − 140·47-s + 4·49-s − 68·53-s + 80·55-s + 136·59-s + 56·67-s + 148·71-s + 27·81-s + 208·89-s + 296·97-s + 120·99-s + 176·103-s + 112·113-s + 208·115-s + 166·121-s − 444·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 4/5·5-s + 2/3·9-s + 1.81·11-s + 2.26·23-s − 3.35·25-s − 0.516·31-s − 1.72·37-s + 8/15·45-s − 2.97·47-s + 4/49·49-s − 1.28·53-s + 1.45·55-s + 2.30·59-s + 0.835·67-s + 2.08·71-s + 1/3·81-s + 2.33·89-s + 3.05·97-s + 1.21·99-s + 1.70·103-s + 0.991·113-s + 1.80·115-s + 1.37·121-s − 3.55·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.211710742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.211710742\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 20 T + 234 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 48 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 4 T^{2} + 3834 T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 100 T^{2} + 50874 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 172 T^{2} - 34650 T^{4} - 172 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 892 T^{2} + 416358 T^{4} - 892 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 26 T + 1200 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 292 T^{2} + 1187046 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 1350 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 32 T + 966 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 3076 T^{2} + 4821894 T^{4} - 3076 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 6844 T^{2} + 18504486 T^{4} - 6844 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 70 T + 3456 T^{2} + 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 34 T + 2640 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 68 T + 6666 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 7636 T^{2} + 33496506 T^{4} - 7636 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 28 T + 6102 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 74 T + 11448 T^{2} - 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 17668 T^{2} + 131640966 T^{4} - 17668 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 16420 T^{2} + 145265274 T^{4} - 16420 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 7780 T^{2} + 34567974 T^{4} - 7780 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 104 T + 15846 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 148 T + 12762 T^{2} - 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.664581310282809923145662041139, −9.257902336745277111481946937430, −9.047457744174211922538504891546, −8.845171506425668050303124580191, −8.411291379245482636091604417415, −8.088609445054731966576989186744, −7.922822951186190320643914487281, −7.32318008379037227210228875706, −7.28731371200311404652080102972, −6.82157515823301350581169958254, −6.62001526323667206060353985539, −6.29141855682245360882675876817, −6.14584226825619226556632129830, −5.51580321004515258046144348974, −5.44751733628282680601678455586, −4.81614586016805839999425900960, −4.77621899189118240297504954448, −4.21607729511304046057854950147, −3.60422289930551220966074034015, −3.47323655569989551423494418876, −3.36556721570737840689086022386, −2.10886718902125671387175263341, −2.03321203986328582004841128023, −1.56872174679718332018763119155, −0.72529323933089635228422231092,
0.72529323933089635228422231092, 1.56872174679718332018763119155, 2.03321203986328582004841128023, 2.10886718902125671387175263341, 3.36556721570737840689086022386, 3.47323655569989551423494418876, 3.60422289930551220966074034015, 4.21607729511304046057854950147, 4.77621899189118240297504954448, 4.81614586016805839999425900960, 5.44751733628282680601678455586, 5.51580321004515258046144348974, 6.14584226825619226556632129830, 6.29141855682245360882675876817, 6.62001526323667206060353985539, 6.82157515823301350581169958254, 7.28731371200311404652080102972, 7.32318008379037227210228875706, 7.922822951186190320643914487281, 8.088609445054731966576989186744, 8.411291379245482636091604417415, 8.845171506425668050303124580191, 9.047457744174211922538504891546, 9.257902336745277111481946937430, 9.664581310282809923145662041139
