| L(s) = 1 | − 104·25-s + 152·49-s + 80·73-s + 400·97-s + 328·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 392·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4.15·25-s + 3.10·49-s + 1.09·73-s + 4.12·97-s + 2.71·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.31·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1824429916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1824429916\) |
| \(L(2)\) |
|
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 | \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 - 18 T + p^{2} T^{2} )^{4}( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 13 | \( ( 1 + 98 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 658 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 478 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 1082 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1862 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 + 578 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 2722 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 2098 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 2882 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 5402 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 6322 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 + 7202 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 2578 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 73 | \( ( 1 - 10 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 - 9542 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 5582 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 + 13282 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 50 T + p^{2} T^{2} )^{8} \) |
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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
−3.51028939852078783971126718601, −3.37152748121130265285008047866, −3.29032718534897641218692282409, −3.28854647334688065658914211987, −3.24115640023573589944094193318, −3.13081110229839771383962954330, −2.63999985615062809980244307809, −2.63590930728273591662771393213, −2.51776447706869197423215308699, −2.34107681220682902516642698752, −2.29774717646677741164165326037, −2.19826644168452973626279813936, −2.03609073331094160235447200360, −1.89020869000223883677406787916, −1.87977628586547739495258776354, −1.76192679234931060741332178620, −1.30159299574085741717607401325, −1.27568988727627622868010805340, −1.26290690739098977798440231437, −1.02746633723292906860017259548, −0.71558993357354787812218665639, −0.55282066410150342486339391362, −0.50145535447357541494564101487, −0.38399446845455750695037565895, −0.02444620261278082565767918755,
0.02444620261278082565767918755, 0.38399446845455750695037565895, 0.50145535447357541494564101487, 0.55282066410150342486339391362, 0.71558993357354787812218665639, 1.02746633723292906860017259548, 1.26290690739098977798440231437, 1.27568988727627622868010805340, 1.30159299574085741717607401325, 1.76192679234931060741332178620, 1.87977628586547739495258776354, 1.89020869000223883677406787916, 2.03609073331094160235447200360, 2.19826644168452973626279813936, 2.29774717646677741164165326037, 2.34107681220682902516642698752, 2.51776447706869197423215308699, 2.63590930728273591662771393213, 2.63999985615062809980244307809, 3.13081110229839771383962954330, 3.24115640023573589944094193318, 3.28854647334688065658914211987, 3.29032718534897641218692282409, 3.37152748121130265285008047866, 3.51028939852078783971126718601
Plot not available for L-functions of degree greater than 10.
