| L(s) = 1 | + 2·4-s + 4·5-s + 16-s − 16·19-s + 8·20-s + 8·25-s − 16·29-s + 16·31-s + 48·41-s + 16·59-s + 24·61-s − 2·64-s − 48·71-s − 32·76-s + 8·79-s + 4·80-s + 18·81-s + 40·89-s − 64·95-s + 16·100-s + 24·101-s + 8·109-s − 32·116-s − 4·121-s + 32·124-s − 8·125-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s + 1.78·5-s + 1/4·16-s − 3.67·19-s + 1.78·20-s + 8/5·25-s − 2.97·29-s + 2.87·31-s + 7.49·41-s + 2.08·59-s + 3.07·61-s − 1/4·64-s − 5.69·71-s − 3.67·76-s + 0.900·79-s + 0.447·80-s + 2·81-s + 4.23·89-s − 6.56·95-s + 8/5·100-s + 2.38·101-s + 0.766·109-s − 2.97·116-s − 0.363·121-s + 2.87·124-s − 0.715·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.78379328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.78379328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 - 4 T + 8 T^{2} + 8 T^{3} - 41 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 + 8 T + 16 T^{2} + 80 T^{3} + 727 T^{4} + 80 p T^{5} + 16 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 36 T^{2} + 298 T^{4} - 2160 T^{6} + 30579 T^{8} - 2160 p^{2} T^{10} + 298 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 8 T + 10 T^{2} + 64 T^{3} - 29 T^{4} + 64 p T^{5} + 10 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 108 T^{2} + 5866 T^{4} + 149040 T^{6} + 2971347 T^{8} + 149040 p^{2} T^{10} + 5866 p^{4} T^{12} + 108 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 92 T^{2} + 4186 T^{4} - 123280 T^{6} - 13364573 T^{8} - 123280 p^{2} T^{10} + 4186 p^{4} T^{12} + 92 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 8 T - 64 T^{2} - 80 T^{3} + 9127 T^{4} - 80 p T^{5} - 64 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T - 8 T^{2} - 360 T^{3} + 10599 T^{4} - 360 p T^{5} - 8 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 236 T^{2} + 31498 T^{4} + 3195440 T^{6} + 260340979 T^{8} + 3195440 p^{2} T^{10} + 31498 p^{4} T^{12} + 236 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 4 T - 122 T^{2} + 80 T^{3} + 11539 T^{4} + 80 p T^{5} - 122 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.78687288580673638851256134378, −4.60503778046362552846181793354, −4.48284620117406212219434165000, −4.30935194868626491306859508180, −4.27628083683920714035866547548, −4.13890220762110812148244348289, −4.09251718762061910817430981414, −3.92974511960388575541158503133, −3.73835134073787942977737456142, −3.51387883453836996596607186572, −3.17526371515837743017447618617, −3.16694948795477962921893291199, −2.86166375789601004525030271409, −2.78507681104172995137174624243, −2.52100673556246555625671644936, −2.51367271070691461862274843814, −2.26348913967598596737749497999, −2.04334240994385253305281612155, −2.03138564069713966821863161942, −1.94679903800092837283671881379, −1.78470312328218601087023553155, −1.34379524314411977246010718772, −0.799310819411581183807673528577, −0.796133082580594783924842149280, −0.63038556313966849172050889117,
0.63038556313966849172050889117, 0.796133082580594783924842149280, 0.799310819411581183807673528577, 1.34379524314411977246010718772, 1.78470312328218601087023553155, 1.94679903800092837283671881379, 2.03138564069713966821863161942, 2.04334240994385253305281612155, 2.26348913967598596737749497999, 2.51367271070691461862274843814, 2.52100673556246555625671644936, 2.78507681104172995137174624243, 2.86166375789601004525030271409, 3.16694948795477962921893291199, 3.17526371515837743017447618617, 3.51387883453836996596607186572, 3.73835134073787942977737456142, 3.92974511960388575541158503133, 4.09251718762061910817430981414, 4.13890220762110812148244348289, 4.27628083683920714035866547548, 4.30935194868626491306859508180, 4.48284620117406212219434165000, 4.60503778046362552846181793354, 4.78687288580673638851256134378
Plot not available for L-functions of degree greater than 10.
