| L(s) = 1 | − 3·2-s + 4·4-s + 2·5-s + 10·7-s − 3·8-s − 6·10-s + 15·11-s − 30·14-s + 16-s + 3·17-s − 9·19-s + 8·20-s − 45·22-s + 25-s + 40·28-s + 15·31-s − 6·32-s − 9·34-s + 20·35-s − 10·37-s + 27·38-s − 6·40-s + 12·41-s + 2·43-s + 60·44-s + 6·47-s + 61·49-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 0.894·5-s + 3.77·7-s − 1.06·8-s − 1.89·10-s + 4.52·11-s − 8.01·14-s + 1/4·16-s + 0.727·17-s − 2.06·19-s + 1.78·20-s − 9.59·22-s + 1/5·25-s + 7.55·28-s + 2.69·31-s − 1.06·32-s − 1.54·34-s + 3.38·35-s − 1.64·37-s + 4.37·38-s − 0.948·40-s + 1.87·41-s + 0.304·43-s + 9.04·44-s + 0.875·47-s + 61/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.078228562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.078228562\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 15 T + 113 T^{2} - 570 T^{3} + 2148 T^{4} - 570 p T^{5} + 113 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3 T - 19 T^{2} + 18 T^{3} + 342 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 9 T + 47 T^{2} + 180 T^{3} + 552 T^{4} + 180 p T^{5} + 47 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 2 T^{2} - 525 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $D_4\times C_2$ | \( 1 + 10 T + 34 T^{2} - 80 T^{3} - 713 T^{4} - 80 p T^{5} + 34 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6 T - 34 T^{2} + 144 T^{3} + 999 T^{4} + 144 p T^{5} - 34 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 27 T + 407 T^{2} - 4428 T^{3} + 36966 T^{4} - 4428 p T^{5} + 407 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 216 p T^{5} - 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 3 T + 101 T^{2} + 294 T^{3} + 6066 T^{4} + 294 p T^{5} + 101 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 7 T - 89 T^{2} + 28 T^{3} + 11272 T^{4} + 28 p T^{5} - 89 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 241 T^{2} + 24396 T^{4} - 241 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 3 T + 125 T^{2} - 366 T^{3} + 9774 T^{4} - 366 p T^{5} + 125 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 50 T^{2} - 80 T^{3} + 8359 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 118 T^{2} - 144 T^{3} + 12591 T^{4} - 144 p T^{5} - 118 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 71 T^{2} + 14064 T^{4} + 71 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.19624672305175991680825524000, −7.11740833908025190970490949751, −6.83158959071798670357593718881, −6.72630873101607530639471116413, −6.30036642390724927834001731999, −6.27191733895882785533636340266, −5.74985857408451047328410474815, −5.72616354126326567747180734938, −5.47843059845758687616743930179, −5.27134016813992585982868998527, −4.71175437891208432213766701651, −4.56094692853818326550636725405, −4.27119389570801657271942475071, −4.09203595925797609969702292634, −4.02502110443418726220031058417, −3.95172286810486995191164728393, −3.19379959257751806837232262211, −2.83936722619259081508749785828, −2.25040523530136082059249618610, −2.10107181951799903151460509764, −1.80274986048979857188851546496, −1.67283805338502136404796949225, −1.08417190685029511028655252669, −1.07080768183246301522022783290, −0.914586153463323266914784643085,
0.914586153463323266914784643085, 1.07080768183246301522022783290, 1.08417190685029511028655252669, 1.67283805338502136404796949225, 1.80274986048979857188851546496, 2.10107181951799903151460509764, 2.25040523530136082059249618610, 2.83936722619259081508749785828, 3.19379959257751806837232262211, 3.95172286810486995191164728393, 4.02502110443418726220031058417, 4.09203595925797609969702292634, 4.27119389570801657271942475071, 4.56094692853818326550636725405, 4.71175437891208432213766701651, 5.27134016813992585982868998527, 5.47843059845758687616743930179, 5.72616354126326567747180734938, 5.74985857408451047328410474815, 6.27191733895882785533636340266, 6.30036642390724927834001731999, 6.72630873101607530639471116413, 6.83158959071798670357593718881, 7.11740833908025190970490949751, 7.19624672305175991680825524000
