| L(s) = 1 | − 9·5-s + 19·7-s + 24·11-s − 61·13-s − 6·17-s − 266·19-s − 69·23-s + 34·25-s + 237·29-s + 211·31-s − 171·35-s + 524·37-s + 468·41-s − 86·43-s − 483·47-s + 540·49-s − 300·53-s − 216·55-s − 168·59-s + 1.04e3·61-s + 549·65-s − 1.16e3·67-s − 624·71-s − 622·73-s + 456·77-s + 349·79-s − 1.22e3·83-s + ⋯ |
| L(s) = 1 | − 0.804·5-s + 1.02·7-s + 0.657·11-s − 1.30·13-s − 0.0856·17-s − 3.21·19-s − 0.625·23-s + 0.271·25-s + 1.51·29-s + 1.22·31-s − 0.825·35-s + 2.32·37-s + 1.78·41-s − 0.304·43-s − 1.49·47-s + 1.57·49-s − 0.777·53-s − 0.529·55-s − 0.370·59-s + 2.20·61-s + 1.04·65-s − 2.12·67-s − 1.04·71-s − 0.997·73-s + 0.674·77-s + 0.497·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5552653953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5552653953\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 9 T + 47 T^{2} - 1944 T^{3} - 24594 T^{4} - 1944 p^{3} T^{5} + 47 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 19 T - 179 T^{2} + 2774 T^{3} + 50128 T^{4} + 2774 p^{3} T^{5} - 179 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 24 T - 1285 T^{2} + 19224 T^{3} + 925104 T^{4} + 19224 p^{3} T^{5} - 1285 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 42334 p^{3} T^{5} - 1367 p^{6} T^{6} + 61 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 9592 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 7 p T + 12234 T^{2} + 7 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3 p T - 20527 T^{2} + 2862 p T^{3} + 433519968 T^{4} + 2862 p^{4} T^{5} - 20527 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 237 T + 4925 T^{2} - 584442 T^{3} + 661218474 T^{4} - 584442 p^{3} T^{5} + 4925 p^{6} T^{6} - 237 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 211 T - 24065 T^{2} - 1899844 T^{3} + 2490210604 T^{4} - 1899844 p^{3} T^{5} - 24065 p^{6} T^{6} - 211 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 262 T + 72162 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 468 T + 27371 T^{2} - 25183548 T^{3} + 16885415064 T^{4} - 25183548 p^{3} T^{5} + 27371 p^{6} T^{6} - 468 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 p T - 17387 T^{2} - 268462 p T^{3} - 6295199732 T^{4} - 268462 p^{4} T^{5} - 17387 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 483 T + 20477 T^{2} + 2495178 T^{3} + 10288968168 T^{4} + 2495178 p^{3} T^{5} + 20477 p^{6} T^{6} + 483 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 150 T + 257074 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 168 T - 388645 T^{2} + 1026648 T^{3} + 125802612624 T^{4} + 1026648 p^{3} T^{5} - 388645 p^{6} T^{6} + 168 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 232819256 p^{3} T^{5} + 424495 p^{6} T^{6} - 1049 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1166 T + 452161 T^{2} + 356643254 T^{3} + 324003162628 T^{4} + 356643254 p^{3} T^{5} + 452161 p^{6} T^{6} + 1166 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 312 T + 498238 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 311 T + 698028 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 349 T - 854801 T^{2} + 3307124 T^{3} + 650611367644 T^{4} + 3307124 p^{3} T^{5} - 854801 p^{6} T^{6} - 349 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1221 T + 78743 T^{2} + 327867804 T^{3} + 814636885368 T^{4} + 327867804 p^{3} T^{5} + 78743 p^{6} T^{6} + 1221 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 492 T + 1092454 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 14111872 p^{3} T^{5} - 1698713 p^{6} T^{6} - 128 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.67814200121087489986769184553, −7.47411448406999928227523360673, −7.15046697327124515909892210835, −6.94590428579461741217179718684, −6.71005421756348397883781264949, −6.16873411202721168436133982951, −6.08937568369012630979169354038, −6.03006674173646252293406128055, −5.92999792085164948497265610221, −5.07734743111835812515919932452, −4.96108656692760530166418605288, −4.61189913945025076161166188374, −4.40744852764220897411737566404, −4.34174240339119656510346987180, −4.27712798732411183148242438876, −3.60845103628456311880285508373, −3.43213155526963916235872301183, −2.91618692246320871626787842321, −2.37885208953177538461643229808, −2.34717366018393227925922344836, −2.24488719250199776581044054414, −1.46466399687772268695736561167, −1.16332368414459574805407525497, −0.70525948237450410749096652644, −0.12961043045441956110420119229,
0.12961043045441956110420119229, 0.70525948237450410749096652644, 1.16332368414459574805407525497, 1.46466399687772268695736561167, 2.24488719250199776581044054414, 2.34717366018393227925922344836, 2.37885208953177538461643229808, 2.91618692246320871626787842321, 3.43213155526963916235872301183, 3.60845103628456311880285508373, 4.27712798732411183148242438876, 4.34174240339119656510346987180, 4.40744852764220897411737566404, 4.61189913945025076161166188374, 4.96108656692760530166418605288, 5.07734743111835812515919932452, 5.92999792085164948497265610221, 6.03006674173646252293406128055, 6.08937568369012630979169354038, 6.16873411202721168436133982951, 6.71005421756348397883781264949, 6.94590428579461741217179718684, 7.15046697327124515909892210835, 7.47411448406999928227523360673, 7.67814200121087489986769184553
