| L(s) = 1 | − 24·11-s + 16·16-s + 100·31-s + 240·41-s − 148·61-s − 528·71-s − 384·101-s − 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 384·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2.18·11-s + 16-s + 3.22·31-s + 5.85·41-s − 2.42·61-s − 7.43·71-s − 3.80·101-s − 1.02·121-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 − 2.18·176-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.483966128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483966128\) |
| \(L(2)\) |
|
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 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 4273 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 39599 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 166942 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 193 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 14882 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1646 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 1382878 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 60 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 5447423 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 8816738 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3737762 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6638 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 18296783 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 132 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 32657282 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 12382 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 94586114 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 137471663 T^{4} + p^{8} T^{8} \) |
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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
−8.958321007283697377248895314114, −8.216763257625378124914596223649, −8.098513317015904372407622648540, −7.952023197328658135539066912349, −7.60121954922206006590605326142, −7.56181991852146259686481541420, −7.18408848523524831620009154164, −7.04346289059040179539233832858, −6.16158494089096047510498295222, −6.10296152920069319670427363293, −6.04219386293400109331606502674, −5.94963851198936088537347123490, −5.21695811026559732085479599226, −5.12807853554647832557193359186, −4.80162278849052333867531690604, −4.20837470804404606862626668567, −4.20246612758976653161383971469, −4.02695098136024601350851357510, −2.98303909394624615090210818392, −2.75857922997806676076361024757, −2.74111739385482804770287422619, −2.59808333354384203725002009685, −1.33275656866598676725437277993, −1.31513936021459234549442556532, −0.35755044013390453147128502600,
0.35755044013390453147128502600, 1.31513936021459234549442556532, 1.33275656866598676725437277993, 2.59808333354384203725002009685, 2.74111739385482804770287422619, 2.75857922997806676076361024757, 2.98303909394624615090210818392, 4.02695098136024601350851357510, 4.20246612758976653161383971469, 4.20837470804404606862626668567, 4.80162278849052333867531690604, 5.12807853554647832557193359186, 5.21695811026559732085479599226, 5.94963851198936088537347123490, 6.04219386293400109331606502674, 6.10296152920069319670427363293, 6.16158494089096047510498295222, 7.04346289059040179539233832858, 7.18408848523524831620009154164, 7.56181991852146259686481541420, 7.60121954922206006590605326142, 7.952023197328658135539066912349, 8.098513317015904372407622648540, 8.216763257625378124914596223649, 8.958321007283697377248895314114
