| L(s) = 1 | − 4·3-s + 10·9-s − 8·11-s + 8·17-s − 20·27-s + 32·33-s + 24·41-s + 16·43-s − 32·51-s − 16·59-s + 32·67-s + 16·73-s + 35·81-s − 24·83-s + 40·89-s + 24·97-s − 80·99-s + 16·107-s + 24·113-s + 12·121-s − 96·123-s + 127-s − 64·129-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s − 2.41·11-s + 1.94·17-s − 3.84·27-s + 5.57·33-s + 3.74·41-s + 2.43·43-s − 4.48·51-s − 2.08·59-s + 3.90·67-s + 1.87·73-s + 35/9·81-s − 2.63·83-s + 4.23·89-s + 2.43·97-s − 8.04·99-s + 1.54·107-s + 2.25·113-s + 1.09·121-s − 8.65·123-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.194243620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.194243620\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 18 T^{4} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 30 T^{4} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 12 T^{2} + 246 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 12 T^{2} + 582 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 1650 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 96 T^{2} + 4098 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 4342 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 108 T^{2} + 6822 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 64 T^{2} + 1234 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 188 T^{2} + 15766 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 172 T^{2} + 15430 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 288 T^{2} + 33090 T^{4} + 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.75450544371424077006569085441, −6.30359560518311708177468082975, −6.21539427976045565535184383833, −5.96060594363477265719184694540, −5.94245407136298232553419302861, −5.58506194691075350789015056474, −5.42883775999320007188950305115, −5.42150204346732117928670016811, −5.06480433673687681357272314377, −4.84641395267085299298420996286, −4.57323082951261740818203677347, −4.50442577350625650473955766344, −4.21176262089256818142144226732, −3.91524021474981203924473416279, −3.46779843672861708262631404616, −3.29479208808159758121386928672, −3.28542176021230469221787547165, −2.54980116666271899808018536456, −2.44903647131749265476563518079, −2.12887893026920818926188999439, −2.07946244841356220580888619823, −1.17914994145276179065768207164, −0.960118775752569016281505089149, −0.802431047703179743830455408225, −0.36521068080376378048145978399,
0.36521068080376378048145978399, 0.802431047703179743830455408225, 0.960118775752569016281505089149, 1.17914994145276179065768207164, 2.07946244841356220580888619823, 2.12887893026920818926188999439, 2.44903647131749265476563518079, 2.54980116666271899808018536456, 3.28542176021230469221787547165, 3.29479208808159758121386928672, 3.46779843672861708262631404616, 3.91524021474981203924473416279, 4.21176262089256818142144226732, 4.50442577350625650473955766344, 4.57323082951261740818203677347, 4.84641395267085299298420996286, 5.06480433673687681357272314377, 5.42150204346732117928670016811, 5.42883775999320007188950305115, 5.58506194691075350789015056474, 5.94245407136298232553419302861, 5.96060594363477265719184694540, 6.21539427976045565535184383833, 6.30359560518311708177468082975, 6.75450544371424077006569085441
