| L(s) = 1 | − 4·5-s − 16·13-s − 4·17-s + 16·23-s + 8·25-s + 4·29-s + 16·31-s + 28·37-s − 12·41-s + 32·47-s − 4·61-s + 64·65-s + 16·71-s + 4·73-s + 16·85-s + 12·97-s − 16·101-s + 32·103-s − 4·109-s + 12·113-s − 64·115-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 4.43·13-s − 0.970·17-s + 3.33·23-s + 8/5·25-s + 0.742·29-s + 2.87·31-s + 4.60·37-s − 1.87·41-s + 4.66·47-s − 0.512·61-s + 7.93·65-s + 1.89·71-s + 0.468·73-s + 1.73·85-s + 1.21·97-s − 1.59·101-s + 3.15·103-s − 0.383·109-s + 1.12·113-s − 5.96·115-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.703750719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.703750719\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 2 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 816 T^{3} + 4418 T^{4} - 816 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 20 T^{3} - 1106 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 944 T^{3} + 6178 T^{4} - 944 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 3668 T^{3} + 25486 T^{4} - 3668 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11830 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 108 T^{3} + 302 T^{4} + 108 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1072 T^{3} + 8962 T^{4} - 1072 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 2402 T^{4} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 20566 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 348 T^{3} - 14194 T^{4} + 348 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.52324521163496317914699553014, −6.19716227264776516878285734771, −5.76678246305103771963960973427, −5.71468770551962822472080152790, −5.62551764084231073157151026687, −4.98643633585998258884808314787, −4.96449295757044350584748943702, −4.78060764379347168060551794763, −4.70033713251472158057506138511, −4.50616022462072209965539410992, −4.37933100594080432931695986650, −4.18679943510782947889201369128, −3.89048454236151551614780440421, −3.50708213471789789302146654810, −3.01010622316719044107653687957, −3.00320468761514185092207286358, −2.93916528870316410948090038500, −2.60013900726021135651132193236, −2.36433755261793288723178151169, −2.11714023246623023451208303636, −2.08070100375707122969714259608, −0.957158360385773751764280080542, −0.940735182566264888276777939461, −0.67952185510918985763698502623, −0.43661528814740993550704076749,
0.43661528814740993550704076749, 0.67952185510918985763698502623, 0.940735182566264888276777939461, 0.957158360385773751764280080542, 2.08070100375707122969714259608, 2.11714023246623023451208303636, 2.36433755261793288723178151169, 2.60013900726021135651132193236, 2.93916528870316410948090038500, 3.00320468761514185092207286358, 3.01010622316719044107653687957, 3.50708213471789789302146654810, 3.89048454236151551614780440421, 4.18679943510782947889201369128, 4.37933100594080432931695986650, 4.50616022462072209965539410992, 4.70033713251472158057506138511, 4.78060764379347168060551794763, 4.96449295757044350584748943702, 4.98643633585998258884808314787, 5.62551764084231073157151026687, 5.71468770551962822472080152790, 5.76678246305103771963960973427, 6.19716227264776516878285734771, 6.52324521163496317914699553014
