| L(s) = 1 | − 2·4-s + 24·13-s + 3·16-s − 16·47-s − 48·52-s − 4·64-s − 48·67-s − 81-s + 8·89-s + 56·101-s + 16·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 308·169-s + 173-s + 179-s + 181-s + 32·188-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4-s + 6.65·13-s + 3/4·16-s − 2.33·47-s − 6.65·52-s − 1/2·64-s − 5.86·67-s − 1/9·81-s + 0.847·89-s + 5.57·101-s + 1.57·103-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 + 23.6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 2.33·188-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.457737255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.457737255\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | | \( 1 \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 + 2 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 958 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \) |
| 31 | $C_2^3$ | \( 1 - 1246 T^{4} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 626 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^3$ | \( 1 + 3794 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 + 1154 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^3$ | \( 1 - 9118 T^{4} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 6146 T^{4} + p^{4} T^{8} \) |
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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
−6.39696254576639840421097014132, −6.37751346644316353341915208050, −6.20650363944950140985794417729, −6.09163585958685105876928427700, −5.76276538350345169889763579322, −5.69248826812880945939162036030, −5.54491818588528339585218700874, −5.15822150412737318690895925776, −4.71968002831761964485574728916, −4.62604203006683068622570878661, −4.33074743475065609988365518699, −4.32562108031942184072451296094, −3.94742485481582007394756314391, −3.56429651247981712878231695176, −3.51693811816339145109280867160, −3.26481196794857030820622080049, −3.21448737995741067852110916425, −3.14981430494682108726412803917, −2.34655988128672462293583320183, −1.92917457264577293618925697938, −1.74716469390857506638793077964, −1.20453553170614642405190814475, −1.18716447005332026001439736456, −1.11855930563323365891349631285, −0.35819779298033902580401367930,
0.35819779298033902580401367930, 1.11855930563323365891349631285, 1.18716447005332026001439736456, 1.20453553170614642405190814475, 1.74716469390857506638793077964, 1.92917457264577293618925697938, 2.34655988128672462293583320183, 3.14981430494682108726412803917, 3.21448737995741067852110916425, 3.26481196794857030820622080049, 3.51693811816339145109280867160, 3.56429651247981712878231695176, 3.94742485481582007394756314391, 4.32562108031942184072451296094, 4.33074743475065609988365518699, 4.62604203006683068622570878661, 4.71968002831761964485574728916, 5.15822150412737318690895925776, 5.54491818588528339585218700874, 5.69248826812880945939162036030, 5.76276538350345169889763579322, 6.09163585958685105876928427700, 6.20650363944950140985794417729, 6.37751346644316353341915208050, 6.39696254576639840421097014132
