| L(s) = 1 | + 72·11-s + 7·16-s + 88·31-s + 72·41-s + 8·61-s − 288·71-s + 432·101-s + 2.75e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 504·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 6.54·11-s + 7/16·16-s + 2.83·31-s + 1.75·41-s + 8/61·61-s − 4.05·71-s + 4.27·101-s + 22.7·121-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 + 0.00578·173-s + 2.86·176-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.971897194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.971897194\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 4702 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 113858 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 475582 T^{4} + p^{8} T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 3168578 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 2956702 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 9478462 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 15243938 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 14394142 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 72 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 10963582 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 7582 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 31395742 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 7742 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 171536062 T^{4} + p^{8} T^{8} \) |
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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
−8.984529558874906902470964877827, −8.496339626870614057560859018177, −8.418108421491999736536416458415, −7.910834889814330713313011947675, −7.51886605476852222920395490747, −7.35933896262784733371094042540, −6.85035907410755275183033147125, −6.75642982065632031280796382958, −6.62139336004109235268204296511, −6.27880921522558439170328361108, −5.98105131061163675952040397280, −5.93270058056036857667692529122, −5.66984499577929574490749411901, −4.64495861664879445859095010401, −4.49705598293395536691563928818, −4.41013004413846891860704902291, −4.28717360520495250577492599402, −3.55284386699416676520756741542, −3.47329177377629077118638421789, −3.40323329891924750603408541975, −2.52225572703026884565144681282, −2.01172045743246777103174722131, −1.32098563901636395253608794760, −1.12239581875821664785303519547, −1.02186076017978405010768415862,
1.02186076017978405010768415862, 1.12239581875821664785303519547, 1.32098563901636395253608794760, 2.01172045743246777103174722131, 2.52225572703026884565144681282, 3.40323329891924750603408541975, 3.47329177377629077118638421789, 3.55284386699416676520756741542, 4.28717360520495250577492599402, 4.41013004413846891860704902291, 4.49705598293395536691563928818, 4.64495861664879445859095010401, 5.66984499577929574490749411901, 5.93270058056036857667692529122, 5.98105131061163675952040397280, 6.27880921522558439170328361108, 6.62139336004109235268204296511, 6.75642982065632031280796382958, 6.85035907410755275183033147125, 7.35933896262784733371094042540, 7.51886605476852222920395490747, 7.910834889814330713313011947675, 8.418108421491999736536416458415, 8.496339626870614057560859018177, 8.984529558874906902470964877827
