| L(s) = 1 | + 3·2-s + 4-s − 6·7-s + 3·8-s + 42·11-s − 78·13-s − 18·14-s + 9·16-s + 102·17-s + 56·19-s + 126·22-s − 48·23-s − 234·26-s − 6·28-s + 318·29-s + 52·31-s − 165·32-s + 306·34-s − 306·37-s + 168·38-s + 408·41-s − 120·43-s + 42·44-s − 144·46-s + 180·47-s − 290·49-s − 78·52-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 0.323·7-s + 0.132·8-s + 1.15·11-s − 1.66·13-s − 0.343·14-s + 9/64·16-s + 1.45·17-s + 0.676·19-s + 1.22·22-s − 0.435·23-s − 1.76·26-s − 0.0404·28-s + 2.03·29-s + 0.301·31-s − 0.911·32-s + 1.54·34-s − 1.35·37-s + 0.717·38-s + 1.55·41-s − 0.425·43-s + 0.143·44-s − 0.461·46-s + 0.558·47-s − 0.845·49-s − 0.208·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.856295751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.856295751\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + p^{3} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 326 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 42 T + 2734 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 p T + 5546 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 p T + 11402 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 56 T + 8598 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 48 T + 24254 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 318 T + 55978 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 52 T + 58782 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 306 T + 94826 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 408 T + 177982 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 120 T + 68150 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 180 T + 128990 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 402 T + 269234 T^{2} - 402 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 732 T + 698582 T^{2} + 732 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 36 T + 384046 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1332 T + 1102034 T^{2} + 1332 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 380 T + 99678 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 984 T + 942182 T^{2} + 984 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1116 T + 1508758 T^{2} + 1116 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 768 T + 1382402 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.94903704569756358426326497767, −11.94346387860994636978927591178, −11.36947303961115371933134026381, −10.36240979570695940946407032855, −10.15541839878537910026852799382, −9.809125067411754901408620420159, −9.074810765468157147871308543604, −8.729182469611316358590114799750, −7.87282757374301840820356457019, −7.42990441439697567149532125474, −6.94352074344351613310366883575, −6.37959091967386491274566000400, −5.55077672619013929452988806514, −5.30829016090893169540249818246, −4.39580327020140988465203035346, −4.29877697518416051588241199469, −3.30170718230716042603967448156, −2.85016827522063193080890320155, −1.72923619680681977535598499642, −0.71535180307929917258243212967,
0.71535180307929917258243212967, 1.72923619680681977535598499642, 2.85016827522063193080890320155, 3.30170718230716042603967448156, 4.29877697518416051588241199469, 4.39580327020140988465203035346, 5.30829016090893169540249818246, 5.55077672619013929452988806514, 6.37959091967386491274566000400, 6.94352074344351613310366883575, 7.42990441439697567149532125474, 7.87282757374301840820356457019, 8.729182469611316358590114799750, 9.074810765468157147871308543604, 9.809125067411754901408620420159, 10.15541839878537910026852799382, 10.36240979570695940946407032855, 11.36947303961115371933134026381, 11.94346387860994636978927591178, 11.94903704569756358426326497767
