| L(s) = 1 | − 2·4-s − 2·5-s + 6·11-s + 6·13-s + 2·17-s + 12·19-s + 4·20-s − 12·23-s + 3·25-s + 6·29-s + 12·31-s − 4·37-s + 4·41-s + 4·43-s − 12·44-s + 6·47-s − 12·52-s − 12·55-s − 4·59-s + 8·64-s − 12·65-s − 8·67-s − 4·68-s + 12·71-s − 24·76-s − 14·79-s − 4·85-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s + 1.80·11-s + 1.66·13-s + 0.485·17-s + 2.75·19-s + 0.894·20-s − 2.50·23-s + 3/5·25-s + 1.11·29-s + 2.15·31-s − 0.657·37-s + 0.624·41-s + 0.609·43-s − 1.80·44-s + 0.875·47-s − 1.66·52-s − 1.61·55-s − 0.520·59-s + 64-s − 1.48·65-s − 0.977·67-s − 0.485·68-s + 1.42·71-s − 2.75·76-s − 1.57·79-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.516067712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.516067712\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 135 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 257 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−9.071909591690618624898709556636, −8.912057149777146714125919779356, −8.379991780454674846004898979313, −8.332979428775281962523358727975, −7.65077403859023603295446257660, −7.48234877377573606479400172701, −7.02394662091311261757133788545, −6.26290181058918537681165676932, −6.04767864297534456620417626782, −6.03792194646420905044200909544, −5.06215173758793164073558378422, −4.81673537060458119819566143057, −4.26429006371015797349625417818, −3.97851752142182424656321860884, −3.45230856729109863583892678141, −3.42771083589053003477445781080, −2.58152376863377329181702339578, −1.66663754915808709059767631528, −0.892568994749630052973776890123, −0.856417933321485659184945988999,
0.856417933321485659184945988999, 0.892568994749630052973776890123, 1.66663754915808709059767631528, 2.58152376863377329181702339578, 3.42771083589053003477445781080, 3.45230856729109863583892678141, 3.97851752142182424656321860884, 4.26429006371015797349625417818, 4.81673537060458119819566143057, 5.06215173758793164073558378422, 6.03792194646420905044200909544, 6.04767864297534456620417626782, 6.26290181058918537681165676932, 7.02394662091311261757133788545, 7.48234877377573606479400172701, 7.65077403859023603295446257660, 8.332979428775281962523358727975, 8.379991780454674846004898979313, 8.912057149777146714125919779356, 9.071909591690618624898709556636
