| L(s) = 1 | + 4·7-s + 20·13-s + 40·19-s + 46·25-s + 76·31-s + 128·37-s − 92·43-s − 78·49-s − 124·61-s − 212·67-s − 208·73-s + 28·79-s + 80·91-s − 28·97-s + 148·103-s − 64·109-s + 394·121-s + 127-s + 131-s + 160·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 4/7·7-s + 1.53·13-s + 2.10·19-s + 1.83·25-s + 2.45·31-s + 3.45·37-s − 2.13·43-s − 1.59·49-s − 2.03·61-s − 3.16·67-s − 2.84·73-s + 0.354·79-s + 0.879·91-s − 0.288·97-s + 1.43·103-s − 0.587·109-s + 3.25·121-s + 0.00787·127-s + 0.00763·131-s + 1.20·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.36720252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.36720252\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 23 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 45 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 394 T^{2} + 66147 T^{4} - 394 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 10 T + 147 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 20 T + 606 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2026 T^{2} + 1583907 T^{4} - 2026 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3166 T^{2} + 3912675 T^{4} - 3166 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 38 T + 1797 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 64 T + 3546 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2782 T^{2} + 4157187 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 46 T + 87 p T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6586 T^{2} + 19745907 T^{4} - 6586 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5194 T^{2} + 14880867 T^{4} - 5194 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 62 T + 6459 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 106 T + 11301 T^{2} + 106 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 11181 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2842 T^{2} + 75503283 T^{4} - 2842 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 14 T + 8283 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.63674795616273895876505659725, −6.42479732401661562307382833439, −6.19606488363671555771599339384, −6.02930158113953161419912389683, −5.96940706798490210695117946167, −5.60357807915222880898435233247, −5.45078866833320293781740234946, −4.94997701080550235673402033966, −4.92721674740632177856721791498, −4.60477279413183678912692375328, −4.42843610949763959023458787702, −4.35137207489972473608390977251, −4.15019956964612132070685409468, −3.47557114106265314696308564854, −3.32062594632506754131328687417, −3.12099422938010203503235940134, −2.94982430615021434405385558993, −2.82858162081145648568382306321, −2.48722870255257886592539478473, −1.71761702231169970825154405321, −1.70575107382543915709252610797, −1.33684776117530933015748656369, −1.12732748443671279187222178068, −0.64340256405058581337037716440, −0.51741568217380455299230901792,
0.51741568217380455299230901792, 0.64340256405058581337037716440, 1.12732748443671279187222178068, 1.33684776117530933015748656369, 1.70575107382543915709252610797, 1.71761702231169970825154405321, 2.48722870255257886592539478473, 2.82858162081145648568382306321, 2.94982430615021434405385558993, 3.12099422938010203503235940134, 3.32062594632506754131328687417, 3.47557114106265314696308564854, 4.15019956964612132070685409468, 4.35137207489972473608390977251, 4.42843610949763959023458787702, 4.60477279413183678912692375328, 4.92721674740632177856721791498, 4.94997701080550235673402033966, 5.45078866833320293781740234946, 5.60357807915222880898435233247, 5.96940706798490210695117946167, 6.02930158113953161419912389683, 6.19606488363671555771599339384, 6.42479732401661562307382833439, 6.63674795616273895876505659725
