| L(s) = 1 | + 4·7-s − 4·9-s − 8·17-s + 16·23-s − 12·25-s − 8·41-s + 32·47-s + 10·49-s − 16·63-s + 32·71-s − 24·73-s + 32·79-s + 2·81-s + 8·89-s − 40·97-s + 32·103-s + 24·113-s − 32·119-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 4/3·9-s − 1.94·17-s + 3.33·23-s − 2.39·25-s − 1.24·41-s + 4.66·47-s + 10/7·49-s − 2.01·63-s + 3.79·71-s − 2.80·73-s + 3.60·79-s + 2/9·81-s + 0.847·89-s − 4.06·97-s + 3.15·103-s + 2.25·113-s − 2.93·119-s − 1.09·121-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 + 2.58·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.030041892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.030041892\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 78 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 150 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 12 T^{2} - 18 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 36 T^{2} + 1038 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 44 T^{2} + 2038 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 116 T^{2} + 5974 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2:C_4$ | \( 1 + 140 T^{2} + 8470 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 84 T^{2} + 5334 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 196 T^{2} + 16174 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 44 T^{2} + 5614 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 204 T^{2} + 18870 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 132 T^{2} + 13134 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−6.75153459971830059266581348867, −6.12508643194114885271545844381, −6.11735198686088139244253116843, −6.07243382107648119891814640649, −5.85826208188774209997500997112, −5.37978435467164555139566370070, −5.16920629400447209468787383992, −5.12684869949443878024881240764, −5.10615350547528799135240786300, −4.70359443630167334754654164063, −4.38090758899624571929091877244, −4.27616477907520258298101433049, −3.92823856200841174389385875988, −3.77566626499754526914794114362, −3.56167506464869717258584949072, −3.16463950107020017705630722470, −2.92532233568599574162167171213, −2.48497696600865014923459916060, −2.44449392282825867472294226931, −2.16536494366441157045555809258, −2.03191357652435797897771359601, −1.43658489468924087175162109080, −1.23925460408635517745640904292, −0.60676135026814819524290227587, −0.50289367066627276125014389252,
0.50289367066627276125014389252, 0.60676135026814819524290227587, 1.23925460408635517745640904292, 1.43658489468924087175162109080, 2.03191357652435797897771359601, 2.16536494366441157045555809258, 2.44449392282825867472294226931, 2.48497696600865014923459916060, 2.92532233568599574162167171213, 3.16463950107020017705630722470, 3.56167506464869717258584949072, 3.77566626499754526914794114362, 3.92823856200841174389385875988, 4.27616477907520258298101433049, 4.38090758899624571929091877244, 4.70359443630167334754654164063, 5.10615350547528799135240786300, 5.12684869949443878024881240764, 5.16920629400447209468787383992, 5.37978435467164555139566370070, 5.85826208188774209997500997112, 6.07243382107648119891814640649, 6.11735198686088139244253116843, 6.12508643194114885271545844381, 6.75153459971830059266581348867
