| L(s) = 1 | − 22·2-s − 2.12e3·4-s + 1.08e4·7-s + 5.90e4·8-s + 3.61e5·11-s + 2.13e6·13-s − 2.39e5·14-s + 1.03e6·16-s − 7.80e6·17-s − 1.55e7·19-s − 7.95e6·22-s − 3.74e7·23-s − 4.69e7·26-s − 2.30e7·28-s + 7.03e7·29-s + 2.98e8·31-s + 1.11e7·32-s + 1.71e8·34-s − 2.36e8·37-s + 3.42e8·38-s + 4.64e8·41-s + 2.42e8·43-s − 7.68e8·44-s + 8.23e8·46-s − 4.37e9·47-s − 1.24e9·49-s − 4.53e9·52-s + ⋯ |
| L(s) = 1 | − 0.486·2-s − 1.03·4-s + 0.244·7-s + 0.637·8-s + 0.677·11-s + 1.59·13-s − 0.118·14-s + 0.247·16-s − 1.33·17-s − 1.44·19-s − 0.329·22-s − 1.21·23-s − 0.774·26-s − 0.253·28-s + 0.636·29-s + 1.87·31-s + 0.0587·32-s + 0.648·34-s − 0.559·37-s + 0.700·38-s + 0.626·41-s + 0.251·43-s − 0.702·44-s + 0.589·46-s − 2.78·47-s − 0.630·49-s − 1.65·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 11 p T + 163 p^{4} T^{2} + 11 p^{12} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 1552 p T + 1364424926 T^{2} - 1552 p^{12} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 361792 T + 429852327334 T^{2} - 361792 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2133732 T + 4716617120926 T^{2} - 2133732 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7804588 T + 76431993653302 T^{2} + 7804588 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 15562224 T + 223670591350918 T^{2} + 15562224 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 37450248 T + 1485695255413774 T^{2} + 37450248 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 70320668 T + 10276443921751438 T^{2} - 70320668 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 298584872 T + 71113481278713662 T^{2} - 298584872 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 236000956 T + 238646163644168046 T^{2} + 236000956 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 464942588 T + 352557098230423222 T^{2} - 464942588 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 242208600 T - 840276877730003210 T^{2} - 242208600 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4375796920 T + 9100022224681989790 T^{2} + 4375796920 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2189541388 T + 18323810419577155774 T^{2} + 2189541388 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5480385856 T + 33382416716089423558 T^{2} - 5480385856 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14557903980 T + \)\(13\!\cdots\!58\)\( T^{2} - 14557903980 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 15918388888 T + \)\(30\!\cdots\!02\)\( T^{2} - 15918388888 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1120561024 T + \)\(32\!\cdots\!86\)\( T^{2} + 1120561024 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24521574348 T + \)\(76\!\cdots\!54\)\( T^{2} - 24521574348 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 79243055560 T + \)\(30\!\cdots\!58\)\( T^{2} + 79243055560 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9245226696 T - \)\(48\!\cdots\!98\)\( T^{2} - 9245226696 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22117321236 T + \)\(55\!\cdots\!78\)\( T^{2} + 22117321236 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 160363673468 T + \)\(20\!\cdots\!62\)\( T^{2} - 160363673468 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−9.810989486178220587193860019242, −9.679146761854877679877327813904, −8.894892052361135479885725831261, −8.522623407166691669922870574525, −8.172244414033224218677561303112, −8.169386338921941075575837188331, −6.81389827082621495829989513269, −6.58341406931125076261160014887, −6.24759340587927086498509349390, −5.51788746319579254897152138903, −4.64682429738633655107635683330, −4.52338267335497871481734234415, −3.88506767514975422265742060684, −3.55398152176002973222593706683, −2.51986381667947543174800103434, −2.06142529694853095886225727192, −1.27822694174096004268704645411, −0.958851288211037846166968390189, 0, 0,
0.958851288211037846166968390189, 1.27822694174096004268704645411, 2.06142529694853095886225727192, 2.51986381667947543174800103434, 3.55398152176002973222593706683, 3.88506767514975422265742060684, 4.52338267335497871481734234415, 4.64682429738633655107635683330, 5.51788746319579254897152138903, 6.24759340587927086498509349390, 6.58341406931125076261160014887, 6.81389827082621495829989513269, 8.169386338921941075575837188331, 8.172244414033224218677561303112, 8.522623407166691669922870574525, 8.894892052361135479885725831261, 9.679146761854877679877327813904, 9.810989486178220587193860019242
