| L(s) = 1 | + 3·9-s + 2·11-s − 12·19-s − 10·29-s + 8·31-s + 12·41-s − 2·49-s − 16·59-s − 44·61-s − 22·79-s − 7·81-s − 4·89-s + 6·99-s + 24·101-s + 26·109-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s − 36·171-s + ⋯ |
| L(s) = 1 | + 9-s + 0.603·11-s − 2.75·19-s − 1.85·29-s + 1.43·31-s + 1.87·41-s − 2/7·49-s − 2.08·59-s − 5.63·61-s − 2.47·79-s − 7/9·81-s − 0.423·89-s + 0.603·99-s + 2.38·101-s + 2.49·109-s − 3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.92·169-s − 2.75·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8319670216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8319670216\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + T^{2} + 64 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 6686 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 139 T^{2} + 8904 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 174 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + 22 T + 226 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 86 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 10150 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 199 T^{2} + 20112 T^{4} - 199 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.14677291695748971781982379078, −6.07951420097898338004145029385, −6.00528596244521783657316911190, −5.78290299963205820760659750433, −5.57682370952244326304818915735, −4.95881985394922716778700282093, −4.91365486849949163037440660511, −4.69953180893672835199418891330, −4.67026291486765874208650768244, −4.28825738439840609679211460628, −4.28508356916486569571280198773, −3.89364590475015411614293452134, −3.82989121372762738932849321672, −3.54492701248456895948612311908, −3.34108839097049890460997584503, −2.78115746327099144425184541953, −2.72375619619036583174805094324, −2.59493010970119243796619690765, −2.33560595878601600038479086338, −1.77410993889226884529692869285, −1.50126008891420000480069824742, −1.46692124714175298088941314771, −1.41616227950890300192401185144, −0.56009966754448093813939260176, −0.16885591377321239388541542978,
0.16885591377321239388541542978, 0.56009966754448093813939260176, 1.41616227950890300192401185144, 1.46692124714175298088941314771, 1.50126008891420000480069824742, 1.77410993889226884529692869285, 2.33560595878601600038479086338, 2.59493010970119243796619690765, 2.72375619619036583174805094324, 2.78115746327099144425184541953, 3.34108839097049890460997584503, 3.54492701248456895948612311908, 3.82989121372762738932849321672, 3.89364590475015411614293452134, 4.28508356916486569571280198773, 4.28825738439840609679211460628, 4.67026291486765874208650768244, 4.69953180893672835199418891330, 4.91365486849949163037440660511, 4.95881985394922716778700282093, 5.57682370952244326304818915735, 5.78290299963205820760659750433, 6.00528596244521783657316911190, 6.07951420097898338004145029385, 6.14677291695748971781982379078
