| L(s) = 1 | + 4·5-s − 2·9-s + 8·19-s + 8·25-s + 8·29-s + 16·31-s − 24·41-s − 8·45-s + 16·49-s − 16·59-s + 24·61-s − 16·79-s + 3·81-s − 32·89-s + 32·95-s − 24·101-s − 24·109-s + 4·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 64·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2/3·9-s + 1.83·19-s + 8/5·25-s + 1.48·29-s + 2.87·31-s − 3.74·41-s − 1.19·45-s + 16/7·49-s − 2.08·59-s + 3.07·61-s − 1.80·79-s + 1/3·81-s − 3.39·89-s + 3.28·95-s − 2.38·101-s − 2.29·109-s + 4/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 5.14·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1718361109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1718361109\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1250 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 6738 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1542 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 276 T^{2} + 32438 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.71615033917889365051271680355, −5.69981809756008738847383342237, −5.66168493836214635226380192685, −5.37436247625350306179736631687, −5.13937244784000361665436975107, −5.13561255662015351067837857613, −4.99450179456941052393743825975, −4.42298613922671301505983586913, −4.39163963409752062781800915277, −4.36172922595048116706620417208, −3.92617311660089698721507968146, −3.74444109631654097981509059312, −3.25779132716824230853886671808, −3.25429604018941805484311855901, −3.06531669101422666275542285758, −2.69200624115798459327207886097, −2.61269823971012573913601631014, −2.49643948346853361644558694936, −2.16291168133642042922214327702, −1.85245462365445737482962103645, −1.41248449617268462384734977436, −1.29585817545417155703813123686, −1.05082387167400933654185443094, −0.928726437205907433432490542839, −0.04578628185703732185850633570,
0.04578628185703732185850633570, 0.928726437205907433432490542839, 1.05082387167400933654185443094, 1.29585817545417155703813123686, 1.41248449617268462384734977436, 1.85245462365445737482962103645, 2.16291168133642042922214327702, 2.49643948346853361644558694936, 2.61269823971012573913601631014, 2.69200624115798459327207886097, 3.06531669101422666275542285758, 3.25429604018941805484311855901, 3.25779132716824230853886671808, 3.74444109631654097981509059312, 3.92617311660089698721507968146, 4.36172922595048116706620417208, 4.39163963409752062781800915277, 4.42298613922671301505983586913, 4.99450179456941052393743825975, 5.13561255662015351067837857613, 5.13937244784000361665436975107, 5.37436247625350306179736631687, 5.66168493836214635226380192685, 5.69981809756008738847383342237, 5.71615033917889365051271680355
