| L(s) = 1 | + 2·2-s + 4-s + 2·5-s − 2·7-s + 4·10-s − 2·11-s + 2·13-s − 4·14-s + 16-s + 4·17-s + 6·19-s + 2·20-s − 4·22-s + 4·23-s + 3·25-s + 4·26-s − 2·28-s − 8·29-s + 12·31-s − 2·32-s + 8·34-s − 4·35-s + 12·38-s − 10·41-s − 2·43-s − 2·44-s + 8·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 1.26·10-s − 0.603·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.970·17-s + 1.37·19-s + 0.447·20-s − 0.852·22-s + 0.834·23-s + 3/5·25-s + 0.784·26-s − 0.377·28-s − 1.48·29-s + 2.15·31-s − 0.353·32-s + 1.37·34-s − 0.676·35-s + 1.94·38-s − 1.56·41-s − 0.304·43-s − 0.301·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.803177036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.803177036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 151 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 121 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 95 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26049845981630323212289996030, −9.791268141694709254995948243740, −9.418679887553456665556912496201, −9.368046954444422935177322869469, −8.440810768690363647788760765134, −8.199459174365219702779933570783, −7.68924573585406078381363536238, −7.08976445875095809099528335959, −6.69514942262324026107291338691, −6.30959450036543717872572883141, −5.61205151606384886952526799177, −5.52166826004903773727527928290, −4.91871515518784025867642200364, −4.83743794234285426491415549875, −3.72785437125309116310886621011, −3.66332065022488016517928301561, −3.05711879703359342182885052726, −2.56236852206653238634458799461, −1.67389539528351470154202294170, −0.873412705238580479693108341594,
0.873412705238580479693108341594, 1.67389539528351470154202294170, 2.56236852206653238634458799461, 3.05711879703359342182885052726, 3.66332065022488016517928301561, 3.72785437125309116310886621011, 4.83743794234285426491415549875, 4.91871515518784025867642200364, 5.52166826004903773727527928290, 5.61205151606384886952526799177, 6.30959450036543717872572883141, 6.69514942262324026107291338691, 7.08976445875095809099528335959, 7.68924573585406078381363536238, 8.199459174365219702779933570783, 8.440810768690363647788760765134, 9.368046954444422935177322869469, 9.418679887553456665556912496201, 9.791268141694709254995948243740, 10.26049845981630323212289996030
