| L(s) = 1 | − 5·3-s − 3·5-s + 9·7-s + 3·9-s − 80·11-s − 26·13-s + 15·15-s + 19·17-s + 84·19-s − 45·21-s − 196·23-s − 239·25-s − 40·27-s − 44·29-s + 86·31-s + 400·33-s − 27·35-s + 209·37-s + 130·39-s − 230·41-s − 287·43-s − 9·45-s − 435·47-s − 111·49-s − 95·51-s − 118·53-s + 240·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.268·5-s + 0.485·7-s + 1/9·9-s − 2.19·11-s − 0.554·13-s + 0.258·15-s + 0.271·17-s + 1.01·19-s − 0.467·21-s − 1.77·23-s − 1.91·25-s − 0.285·27-s − 0.281·29-s + 0.498·31-s + 2.11·33-s − 0.130·35-s + 0.928·37-s + 0.533·39-s − 0.876·41-s − 1.01·43-s − 0.0298·45-s − 1.35·47-s − 0.323·49-s − 0.260·51-s − 0.305·53-s + 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 248 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 192 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 80 T + 3650 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 19 T + 8688 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 84 T + 11130 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 196 T + 33326 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 44 T + 10094 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 86 T + 56518 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 209 T + 112120 T^{2} - 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 287 T + 92698 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 435 T + 192728 T^{2} + 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 118 T + 297410 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 368 T + 379266 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 68 T + 373930 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 131 T + 493328 T^{2} - 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1008 T + 1233294 T^{2} - 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1958 T + 1961238 T^{2} + 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 928 T + 943870 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.51533502890607168260277187719, −11.49633744174818056986237180542, −10.78170541356509805002664158193, −10.18527708109680135632618275833, −9.839457269766339393644111305126, −9.601202423789449997924744423882, −8.334562551605682742618043101293, −8.141650479274049139063362431739, −7.61579693765313056237431763556, −7.35811872376366126176835721091, −6.21767943204107209004571371958, −5.94209012268120497054491524259, −5.17398079438253441113167640479, −5.07030050082227723364606538447, −4.18380560692029457266100247885, −3.33449790633096667866136368528, −2.48470255773777236856899999360, −1.66305058468006764603969257999, 0, 0,
1.66305058468006764603969257999, 2.48470255773777236856899999360, 3.33449790633096667866136368528, 4.18380560692029457266100247885, 5.07030050082227723364606538447, 5.17398079438253441113167640479, 5.94209012268120497054491524259, 6.21767943204107209004571371958, 7.35811872376366126176835721091, 7.61579693765313056237431763556, 8.141650479274049139063362431739, 8.334562551605682742618043101293, 9.601202423789449997924744423882, 9.839457269766339393644111305126, 10.18527708109680135632618275833, 10.78170541356509805002664158193, 11.49633744174818056986237180542, 11.51533502890607168260277187719
