| L(s) = 1 | + 4·5-s + 2·9-s + 16·19-s + 2·25-s + 24·29-s + 8·45-s − 12·49-s − 64·71-s − 5·81-s + 64·95-s − 24·101-s + 28·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 96·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 32·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2/3·9-s + 3.67·19-s + 2/5·25-s + 4.45·29-s + 1.19·45-s − 1.71·49-s − 7.59·71-s − 5/9·81-s + 6.56·95-s − 2.38·101-s + 2.54·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.97·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.76·169-s + 2.44·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.015856388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.015856388\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−6.44375510679436736851278288434, −6.23071016155364412255598996304, −6.14744258181104400813922009545, −5.82542803974081322852340204214, −5.74113501199450457036306765535, −5.65766731374019834445471335887, −5.22696226806443323641279812631, −5.14832446649536574811105896226, −4.72830781021391084594304914983, −4.66357949689840919655844865714, −4.47484426320463717101220551369, −4.40687035537148841549816319014, −3.90693112626466777532961724158, −3.55568948059686812968872063106, −3.33363529912541456609811692173, −3.05016504194473706288263644985, −2.82257430312109484057868915806, −2.71277352617491992824383868754, −2.65526569882478860192974647006, −1.99587611241436700614317676610, −1.60197242242616415900737654005, −1.40135360115666312347419510945, −1.38464148445418872874196380401, −0.974724716948452816681226046495, −0.45584086118749131635439930236,
0.45584086118749131635439930236, 0.974724716948452816681226046495, 1.38464148445418872874196380401, 1.40135360115666312347419510945, 1.60197242242616415900737654005, 1.99587611241436700614317676610, 2.65526569882478860192974647006, 2.71277352617491992824383868754, 2.82257430312109484057868915806, 3.05016504194473706288263644985, 3.33363529912541456609811692173, 3.55568948059686812968872063106, 3.90693112626466777532961724158, 4.40687035537148841549816319014, 4.47484426320463717101220551369, 4.66357949689840919655844865714, 4.72830781021391084594304914983, 5.14832446649536574811105896226, 5.22696226806443323641279812631, 5.65766731374019834445471335887, 5.74113501199450457036306765535, 5.82542803974081322852340204214, 6.14744258181104400813922009545, 6.23071016155364412255598996304, 6.44375510679436736851278288434
