| L(s) = 1 | − 4·7-s − 2·9-s + 16·17-s + 8·23-s − 4·25-s + 16·31-s − 32·41-s − 16·47-s + 10·49-s + 8·63-s + 24·71-s − 24·73-s + 16·79-s + 3·81-s − 8·97-s − 16·103-s − 24·113-s − 64·119-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 − 2/3·9-s + 3.88·17-s + 1.66·23-s − 4/5·25-s + 2.87·31-s − 4.99·41-s − 2.33·47-s + 10/7·49-s + 1.00·63-s + 2.84·71-s − 2.80·73-s + 1.80·79-s + 1/3·81-s − 0.812·97-s − 1.57·103-s − 2.25·113-s − 5.86·119-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427752163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427752163\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 950 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 5290 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.70496765449992146082840715191, −5.56766786404473278698293926236, −5.44368171977055426654156620531, −5.11405439469484793621117783768, −4.99119089105707451068748336373, −4.92622843749776490052142855343, −4.87856517291846555325215162217, −4.27465764795217430940103330284, −4.16560479525379401625401012912, −4.03077924771426807265346827817, −3.58526702375783077420775090487, −3.38456732783657586163937689122, −3.29423491132766081156188884783, −3.24873617873961293269249022173, −3.09028291474954499033122329197, −2.83159168981987187148819624927, −2.75673336249064848369823647687, −2.18094621976099529160313345998, −2.09873858791660656118434884941, −1.63695664133786720557699602520, −1.46370375224168878393165729859, −1.08050424058809366856112584876, −1.05063156098181006688982656992, −0.53320115463862594253325467587, −0.25556661915958357521208178094,
0.25556661915958357521208178094, 0.53320115463862594253325467587, 1.05063156098181006688982656992, 1.08050424058809366856112584876, 1.46370375224168878393165729859, 1.63695664133786720557699602520, 2.09873858791660656118434884941, 2.18094621976099529160313345998, 2.75673336249064848369823647687, 2.83159168981987187148819624927, 3.09028291474954499033122329197, 3.24873617873961293269249022173, 3.29423491132766081156188884783, 3.38456732783657586163937689122, 3.58526702375783077420775090487, 4.03077924771426807265346827817, 4.16560479525379401625401012912, 4.27465764795217430940103330284, 4.87856517291846555325215162217, 4.92622843749776490052142855343, 4.99119089105707451068748336373, 5.11405439469484793621117783768, 5.44368171977055426654156620531, 5.56766786404473278698293926236, 5.70496765449992146082840715191
