| L(s) = 1 | + 16·13-s + 20·25-s + 28·49-s − 32·61-s − 24·73-s − 40·97-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 136·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 4.43·13-s + 4·25-s + 4·49-s − 4.09·61-s − 2.80·73-s − 4.06·97-s − 4·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 10.4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.96975914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.96975914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 2 p T^{2} + 51 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 2 p T^{2} + 123 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 p T^{2} + 267 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 28 T^{2} + 342 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 p T^{2} + 2283 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 52 T^{2} + 2502 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 100 T^{2} + 5382 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 26 T^{2} + 387 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 148 T^{2} + 10902 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 20 T^{2} - 522 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 52 T^{2} + 7302 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 74 T^{2} + 12747 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 188 T^{2} + 21222 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 5 T + p T^{2} )^{8} \) |
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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
−4.04001557932732093333998421094, −3.65989893594197516045266365702, −3.64124188422156186342450232739, −3.52789201108675002635632690170, −3.45398995811420679498994638287, −3.44895974384156035646219924001, −3.13933296115095767517559616887, −2.99964976748049310157864451397, −2.90380233151201926832612974100, −2.73834587221283664102012434564, −2.70330996327922190236738881521, −2.65466276443996398665011776040, −2.58924248601055898056468370426, −2.18495096536102471051271913203, −1.97001597124107924997606638913, −1.96958759958764648889852690180, −1.49626030450248948723206705164, −1.38662198568049194703924992363, −1.37706470852213439792547562588, −1.33382656468913062578702097741, −1.33327382355754718703690081059, −0.996301559503478775598387172555, −0.56770975674730254520479892569, −0.53299476197290196490512769634, −0.43539434004673015020903518404,
0.43539434004673015020903518404, 0.53299476197290196490512769634, 0.56770975674730254520479892569, 0.996301559503478775598387172555, 1.33327382355754718703690081059, 1.33382656468913062578702097741, 1.37706470852213439792547562588, 1.38662198568049194703924992363, 1.49626030450248948723206705164, 1.96958759958764648889852690180, 1.97001597124107924997606638913, 2.18495096536102471051271913203, 2.58924248601055898056468370426, 2.65466276443996398665011776040, 2.70330996327922190236738881521, 2.73834587221283664102012434564, 2.90380233151201926832612974100, 2.99964976748049310157864451397, 3.13933296115095767517559616887, 3.44895974384156035646219924001, 3.45398995811420679498994638287, 3.52789201108675002635632690170, 3.64124188422156186342450232739, 3.65989893594197516045266365702, 4.04001557932732093333998421094
Plot not available for L-functions of degree greater than 10.
