| L(s) = 1 | − 5·3-s + 9·4-s + 4·7-s + 9·9-s − 45·12-s − 8·13-s + 37·16-s − 36·19-s − 20·21-s + 93·25-s − 10·27-s + 36·28-s + 46·31-s + 81·36-s − 90·37-s + 40·39-s + 96·43-s − 185·48-s − 120·49-s − 72·52-s + 180·57-s − 24·61-s + 36·63-s + 81·64-s − 58·67-s − 284·73-s − 465·75-s + ⋯ |
| L(s) = 1 | − 5/3·3-s + 9/4·4-s + 4/7·7-s + 9-s − 3.75·12-s − 0.615·13-s + 2.31·16-s − 1.89·19-s − 0.952·21-s + 3.71·25-s − 0.370·27-s + 9/7·28-s + 1.48·31-s + 9/4·36-s − 2.43·37-s + 1.02·39-s + 2.23·43-s − 3.85·48-s − 2.44·49-s − 1.38·52-s + 3.15·57-s − 0.393·61-s + 4/7·63-s + 1.26·64-s − 0.865·67-s − 3.89·73-s − 6.19·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9408209861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9408209861\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 5 T + 16 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 9 T^{2} + 11 p^{2} T^{4} - 9 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 93 T^{2} + 3404 T^{4} - 93 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 66 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 210 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 216 T^{2} + 149006 T^{4} + 216 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 18 T + 506 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 477 T^{2} + 607580 T^{4} - 477 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 188 T^{2} + 246 p^{2} T^{4} + 188 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 23 T + 1848 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 45 T + 2840 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5460 T^{2} + 13000934 T^{4} - 5460 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 48 T + 4142 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6120 T^{2} + 18979214 T^{4} - 6120 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3112 T^{2} + 2490798 T^{4} - 3112 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9921 T^{2} + 45659744 T^{4} - 9921 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 4178 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 29 T + 6804 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 19933 T^{2} + 150145500 T^{4} - 19933 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 142 T + 15402 T^{2} + 142 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 38 T + 7266 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 22120 T^{2} + 214834542 T^{4} - 22120 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 10897 T^{2} + 51258576 T^{4} - 10897 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 109 T + 132 p T^{2} - 109 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.38141397789172462770821812042, −11.74688001601902710871222810101, −11.70968852981559669781787698033, −11.51813828637949980142045169356, −11.00486444852050747712896245855, −10.84036526998441555175128507846, −10.56179720247796994474597070107, −10.34274441959044053426000489874, −10.16038445364951601916138645488, −9.244205530794933589757914979846, −8.918944740889346157887122516605, −8.429983005306895403306275961389, −8.251674173479031250181188277524, −7.37441175036494008532991693648, −7.15675360019047296817432946775, −6.98003003155221187853410198895, −6.46829480048815188021083183211, −6.06958190876563003013349839308, −6.02035549250298108280615147416, −5.15317017260276807481651970301, −4.76262208178670543919733466368, −4.45329971210931960882440347768, −3.09730072256132962541140336218, −2.65193927166806668305012795342, −1.65654041489718236067155511163,
1.65654041489718236067155511163, 2.65193927166806668305012795342, 3.09730072256132962541140336218, 4.45329971210931960882440347768, 4.76262208178670543919733466368, 5.15317017260276807481651970301, 6.02035549250298108280615147416, 6.06958190876563003013349839308, 6.46829480048815188021083183211, 6.98003003155221187853410198895, 7.15675360019047296817432946775, 7.37441175036494008532991693648, 8.251674173479031250181188277524, 8.429983005306895403306275961389, 8.918944740889346157887122516605, 9.244205530794933589757914979846, 10.16038445364951601916138645488, 10.34274441959044053426000489874, 10.56179720247796994474597070107, 10.84036526998441555175128507846, 11.00486444852050747712896245855, 11.51813828637949980142045169356, 11.70968852981559669781787698033, 11.74688001601902710871222810101, 12.38141397789172462770821812042
