| L(s) = 1 | − 2·9-s + 4·11-s + 2·19-s − 12·29-s − 4·31-s + 26·41-s + 19·49-s + 18·59-s − 36·61-s − 2·71-s + 20·79-s + 3·81-s + 28·89-s − 8·99-s − 10·101-s + 46·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·171-s + 173-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 1.20·11-s + 0.458·19-s − 2.22·29-s − 0.718·31-s + 4.06·41-s + 19/7·49-s + 2.34·59-s − 4.60·61-s − 0.237·71-s + 2.25·79-s + 1/3·81-s + 2.96·89-s − 0.804·99-s − 0.995·101-s + 4.40·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.305·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.581145617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.581145617\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 19 T^{2} + 184 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 13 T + 120 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3934 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 120 T^{2} + 7406 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 32 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 18 T + 186 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 30 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 296 T^{2} + 35614 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 63 T^{2} - 4096 T^{4} - 63 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.56584741251328464633339097460, −5.13543245277513539549058082005, −5.06482023888094300034444304260, −4.96710265043942452487736436154, −4.63257169078942759781161706908, −4.33346452354053314409993809993, −4.24513693992092154097054843112, −4.14018042449451032261335185468, −4.09679227351756139955267571161, −3.60407855272112827393441716915, −3.51673729679366771280514584135, −3.39179766600735901368669616931, −3.33314919844553315817816378143, −2.88230797552449687550780316320, −2.71315176552666879473296252362, −2.45809023547575503684062152998, −2.39292730214955445939893480765, −1.93553501828033173505865182517, −1.90227338634737567905977426659, −1.75585993872063300135123715443, −1.39562000113867900699131703535, −0.914484491811243521880105363400, −0.75728383010763361591066210110, −0.69830031549275375662024670473, −0.30970527821904697073224615829,
0.30970527821904697073224615829, 0.69830031549275375662024670473, 0.75728383010763361591066210110, 0.914484491811243521880105363400, 1.39562000113867900699131703535, 1.75585993872063300135123715443, 1.90227338634737567905977426659, 1.93553501828033173505865182517, 2.39292730214955445939893480765, 2.45809023547575503684062152998, 2.71315176552666879473296252362, 2.88230797552449687550780316320, 3.33314919844553315817816378143, 3.39179766600735901368669616931, 3.51673729679366771280514584135, 3.60407855272112827393441716915, 4.09679227351756139955267571161, 4.14018042449451032261335185468, 4.24513693992092154097054843112, 4.33346452354053314409993809993, 4.63257169078942759781161706908, 4.96710265043942452487736436154, 5.06482023888094300034444304260, 5.13543245277513539549058082005, 5.56584741251328464633339097460
