| L(s) = 1 | + 4·4-s − 20·16-s − 44·25-s + 148·31-s + 244·37-s + 98·49-s − 160·64-s − 164·67-s + 716·97-s − 176·100-s − 580·103-s + 592·124-s + 127-s + 131-s + 137-s + 139-s + 976·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 652·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s − 5/4·16-s − 1.75·25-s + 4.77·31-s + 6.59·37-s + 2·49-s − 5/2·64-s − 2.44·67-s + 7.38·97-s − 1.75·100-s − 5.63·103-s + 4.77·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 6.59·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.85·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.174299492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.174299492\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 326 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 719 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 1040 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 668 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 1628 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 1670 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3536 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5600 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6512 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7199 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 41 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 4250 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6983 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 4393 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4652 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 13250 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 179 T + p^{2} T^{2} )^{4} \) |
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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
−6.83626218952415708561079916096, −6.40196867919312116501040250305, −6.25002150336059349826707155461, −6.19462194545537248973332205770, −6.12549024647531104969843815081, −6.10399177553666725344375529746, −5.62892683383876979634473968223, −5.26386448730286087545495837195, −4.95456021599489973787343263596, −4.66199673925775338137865842066, −4.36988933134928217750496930335, −4.33379411960226507262428629655, −4.29248496859419493184283375350, −3.99918897239220515983959539854, −3.38043228002419989085320720770, −3.17289522819748914145323946618, −2.86427358972875563861654681559, −2.48122632029138563859209756430, −2.44488727613109945546662741403, −2.31924535106109136497345140051, −2.04872473609516542925899274036, −1.06899049116884965222203594917, −1.06685670590079220959443762609, −1.04793862924606999320152067388, −0.22992528383243429866325596199,
0.22992528383243429866325596199, 1.04793862924606999320152067388, 1.06685670590079220959443762609, 1.06899049116884965222203594917, 2.04872473609516542925899274036, 2.31924535106109136497345140051, 2.44488727613109945546662741403, 2.48122632029138563859209756430, 2.86427358972875563861654681559, 3.17289522819748914145323946618, 3.38043228002419989085320720770, 3.99918897239220515983959539854, 4.29248496859419493184283375350, 4.33379411960226507262428629655, 4.36988933134928217750496930335, 4.66199673925775338137865842066, 4.95456021599489973787343263596, 5.26386448730286087545495837195, 5.62892683383876979634473968223, 6.10399177553666725344375529746, 6.12549024647531104969843815081, 6.19462194545537248973332205770, 6.25002150336059349826707155461, 6.40196867919312116501040250305, 6.83626218952415708561079916096
