| L(s) = 1 | − 2·3-s − 2·5-s + 6·7-s + 3·9-s − 2·13-s + 4·15-s + 2·19-s − 12·21-s + 4·23-s − 2·25-s − 4·27-s − 8·29-s + 14·31-s − 12·35-s + 4·39-s − 14·41-s + 4·43-s − 6·45-s + 8·47-s + 18·49-s − 8·53-s − 4·57-s + 4·59-s + 16·61-s + 18·63-s + 4·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 2.26·7-s + 9-s − 0.554·13-s + 1.03·15-s + 0.458·19-s − 2.61·21-s + 0.834·23-s − 2/5·25-s − 0.769·27-s − 1.48·29-s + 2.51·31-s − 2.02·35-s + 0.640·39-s − 2.18·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s + 18/7·49-s − 1.09·53-s − 0.529·57-s + 0.520·59-s + 2.04·61-s + 2.26·63-s + 0.496·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.556367111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556367111\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 14 T + 106 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−9.834096414242145080852646350180, −9.801953882476658130959436027994, −8.874472604324464492396088391494, −8.691889966785871100084168311485, −8.021488268728704913289073167464, −7.965104244760045909481337081982, −7.33369870295830015764677198391, −7.30930144294875900153204955992, −6.54009852843961016193061935405, −6.21249569375365825074518450026, −5.47549923340203215077026174519, −5.15979150579379746385401502255, −4.75982102718565161094683902735, −4.68390701152853622123710499884, −3.80662275402744321223719606504, −3.64692729433716486070657797497, −2.53443286794994088595005391960, −2.00782760599828459918974298646, −1.27538981329969259346157492479, −0.64451191013089312161505119599,
0.64451191013089312161505119599, 1.27538981329969259346157492479, 2.00782760599828459918974298646, 2.53443286794994088595005391960, 3.64692729433716486070657797497, 3.80662275402744321223719606504, 4.68390701152853622123710499884, 4.75982102718565161094683902735, 5.15979150579379746385401502255, 5.47549923340203215077026174519, 6.21249569375365825074518450026, 6.54009852843961016193061935405, 7.30930144294875900153204955992, 7.33369870295830015764677198391, 7.965104244760045909481337081982, 8.021488268728704913289073167464, 8.691889966785871100084168311485, 8.874472604324464492396088391494, 9.801953882476658130959436027994, 9.834096414242145080852646350180
