| L(s) = 1 | + 6·9-s − 72·17-s + 68·25-s + 72·41-s + 100·49-s − 328·73-s + 27·81-s + 504·89-s + 440·97-s − 504·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 432·153-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 4.23·17-s + 2.71·25-s + 1.75·41-s + 2.04·49-s − 4.49·73-s + 1/3·81-s + 5.66·89-s + 4.53·97-s − 4.46·113-s − 3.20·121-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 − 2.82·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.877496290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.877496290\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 670 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1666 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 430 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2446 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2690 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3682 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3074 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2258 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 8546 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8354 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 8594 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 3550 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{4} \) |
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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
−7.83467195774336091034254854193, −7.66498121682739745034298865301, −7.56553605493569526247158035262, −7.12833485260679962922957985679, −7.04548469267014271353779973728, −6.68087479353083523481473786644, −6.45622806814619324450059124112, −6.27392022834307555599641346769, −6.25609057044065710435807964904, −5.68459333959429084492062827074, −5.33239788539587570502146866863, −4.96186869166213398999507704602, −4.84188599838223589715607340585, −4.55049271045132023628077433739, −4.22003201048297053337997931721, −4.09998419567446782834805318384, −3.89001077513743863548863622670, −3.10915964936445703698215728781, −3.02454273033745894996260135960, −2.45595250211152728058045069491, −2.36685497314927407665575740552, −1.95232507248799730840320909591, −1.46485556768182803787052876897, −0.78042646156902315649360551355, −0.44727877567296724055635898644,
0.44727877567296724055635898644, 0.78042646156902315649360551355, 1.46485556768182803787052876897, 1.95232507248799730840320909591, 2.36685497314927407665575740552, 2.45595250211152728058045069491, 3.02454273033745894996260135960, 3.10915964936445703698215728781, 3.89001077513743863548863622670, 4.09998419567446782834805318384, 4.22003201048297053337997931721, 4.55049271045132023628077433739, 4.84188599838223589715607340585, 4.96186869166213398999507704602, 5.33239788539587570502146866863, 5.68459333959429084492062827074, 6.25609057044065710435807964904, 6.27392022834307555599641346769, 6.45622806814619324450059124112, 6.68087479353083523481473786644, 7.04548469267014271353779973728, 7.12833485260679962922957985679, 7.56553605493569526247158035262, 7.66498121682739745034298865301, 7.83467195774336091034254854193
