| L(s) = 1 | + 2·5-s + 2·11-s + 2·17-s − 6·19-s − 2·23-s + 25-s − 20·29-s − 2·31-s + 4·37-s + 12·41-s + 8·43-s + 12·47-s − 7·49-s − 4·53-s + 4·55-s + 6·59-s − 16·61-s + 4·67-s − 44·71-s + 6·79-s + 12·83-s + 4·85-s − 2·89-s − 12·95-s + 32·97-s + 6·101-s − 24·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.603·11-s + 0.485·17-s − 1.37·19-s − 0.417·23-s + 1/5·25-s − 3.71·29-s − 0.359·31-s + 0.657·37-s + 1.87·41-s + 1.21·43-s + 1.75·47-s − 49-s − 0.549·53-s + 0.539·55-s + 0.781·59-s − 2.04·61-s + 0.488·67-s − 5.22·71-s + 0.675·79-s + 1.31·83-s + 0.433·85-s − 0.211·89-s − 1.23·95-s + 3.24·97-s + 0.597·101-s − 2.36·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5656197542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5656197542\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 12 p T^{5} - 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 12 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 6 p T^{3} - 36 p T^{4} - 6 p^{2} T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 76 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 2 T - p T^{2} - 54 T^{3} + 140 T^{4} - 54 p T^{5} - p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 55 T^{2} + 12 T^{3} + 3080 T^{4} + 12 p T^{5} - 55 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T - 66 T^{2} - 96 T^{3} + 3067 T^{4} - 96 p T^{5} - 66 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 28 T^{2} + 324 T^{3} - 1509 T^{4} + 324 p T^{5} - 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 115 T^{2} + 12 T^{3} + 11600 T^{4} + 12 p T^{5} - 115 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 22 T + 256 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 103 T^{2} + 114 T^{3} + 10236 T^{4} + 114 p T^{5} - 103 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 2 T - 348 T^{3} - 8261 T^{4} - 348 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−6.99055892359695165805950821657, −6.61889529485795986571443313741, −6.25347578913483579641513818144, −6.15674738943037634945185639912, −6.10183484186941932344513029080, −5.85417019853883451977584433147, −5.79417144096410689711830311289, −5.44019881541301818436715284472, −5.13700009680785864729033428788, −4.98300139744358199608984380729, −4.65858099046749551853078638757, −4.28497816259393512866648339923, −4.16038531675518489587041313879, −3.95871108753831080156254824841, −3.79662224238735448461526805623, −3.48873397837469220704955005331, −3.04959555633982764802269314723, −2.84095122546039708125001086604, −2.57417255265591881432418633395, −2.09667361890457538878958205230, −1.99080297305571502335173203087, −1.79725246722893934108647973852, −1.22800476100533845974334487086, −1.05728629360491220302824439036, −0.14719914391265590872760251820,
0.14719914391265590872760251820, 1.05728629360491220302824439036, 1.22800476100533845974334487086, 1.79725246722893934108647973852, 1.99080297305571502335173203087, 2.09667361890457538878958205230, 2.57417255265591881432418633395, 2.84095122546039708125001086604, 3.04959555633982764802269314723, 3.48873397837469220704955005331, 3.79662224238735448461526805623, 3.95871108753831080156254824841, 4.16038531675518489587041313879, 4.28497816259393512866648339923, 4.65858099046749551853078638757, 4.98300139744358199608984380729, 5.13700009680785864729033428788, 5.44019881541301818436715284472, 5.79417144096410689711830311289, 5.85417019853883451977584433147, 6.10183484186941932344513029080, 6.15674738943037634945185639912, 6.25347578913483579641513818144, 6.61889529485795986571443313741, 6.99055892359695165805950821657
