L(s) = 1 | − 4·8-s − 4·13-s + 3·16-s − 20·17-s − 10·25-s + 4·32-s + 4·37-s + 16·41-s + 4·53-s − 32·61-s + 8·64-s + 44·73-s − 3·81-s − 20·97-s − 40·101-s + 16·104-s − 52·113-s + 60·121-s − 16·125-s + 127-s − 16·128-s + 131-s + 80·136-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1.41·8-s − 1.10·13-s + 3/4·16-s − 4.85·17-s − 2·25-s + 0.707·32-s + 0.657·37-s + 2.49·41-s + 0.549·53-s − 4.09·61-s + 64-s + 5.14·73-s − 1/3·81-s − 2.03·97-s − 3.98·101-s + 1.56·104-s − 4.89·113-s + 5.45·121-s − 1.43·125-s + 0.0887·127-s − 1.41·128-s + 0.0873·131-s + 6.85·136-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
Λ(s)=(=((224⋅312⋅512)s/2ΓC(s)12L(s)Λ(2−s)
Λ(s)=(=((224⋅312⋅512)s/2ΓC(s+1/2)12L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.1709821360 |
L(21) |
≈ |
0.1709821360 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+p2T3−3T4−p2T5+p3T6−p3T7−3p2T8+p5T9+p6T12 |
| 3 | (1+T4)3 |
| 5 | (1+pT2+8T3+p2T4+p3T6)2 |
good | 7 | 1−66T4+671T8+41476T12+671p4T16−66p8T20+p12T24 |
| 11 | (1−30T2+491T4−6076T6+491p2T8−30p4T10+p6T12)2 |
| 13 | (1+2T+2T2−6T3+27T4+580T5+1124T6+580pT7+27p2T8−6p3T9+2p4T10+2p5T11+p6T12)2 |
| 17 | (1+10T+50T2+250T3+1155T4+4260T5+16100T6+4260pT7+1155p2T8+250p3T9+50p4T10+10p5T11+p6T12)2 |
| 19 | (1+74T2+2775T4+65228T6+2775p2T8+74p4T10+p6T12)2 |
| 23 | 1+1190T4+607727T8+223349076T12+607727p4T16+1190p8T20+p12T24 |
| 29 | (1−154T2+10395T4−392596T6+10395p2T8−154p4T10+p6T12)2 |
| 31 | (1−74T2+4175T4−143436T6+4175p2T8−74p4T10+p6T12)2 |
| 37 | (1−10T+93T2−472T3+93pT4−10p2T5+p3T6)2(1+8T−11T2−424T3−11pT4+8p2T5+p3T6)2 |
| 41 | (1−4T+103T2−264T3+103pT4−4p2T5+p3T6)4 |
| 43 | 1−6826T4+23542879T8−52748850124T12+23542879p4T16−6826p8T20+p12T24 |
| 47 | 1+1606T4+1555663T8−11534742380T12+1555663p4T16+1606p8T20+p12T24 |
| 53 | (1−2T+2T2−90T3+5291T4−7252T5+7972T6−7252pT7+5291p2T8−90p3T9+2p4T10−2p5T11+p6T12)2 |
| 59 | (1+254T2+30475T4+2237948T6+30475p2T8+254p4T10+p6T12)2 |
| 61 | (1+8T+83T2+1152T3+83pT4+8p2T5+p3T6)4 |
| 67 | 1−6346T4+60212671T8−234156159244T12+60212671p4T16−6346p8T20+p12T24 |
| 71 | (1−170T2+18527T4−1427916T6+18527p2T8−170p4T10+p6T12)2 |
| 73 | (1−22T+242T2−2246T3+20271T4−179604T5+1567964T6−179604pT7+20271p2T8−2246p3T9+242p4T10−22p5T11+p6T12)2 |
| 79 | (1+170T2+10415T4+507660T6+10415p2T8+170p4T10+p6T12)2 |
| 83 | 1+7542T4−16589569T8−295904039564T12−16589569p4T16+7542p8T20+p12T24 |
| 89 | (1−462T2+94223T4−10861604T6+94223p2T8−462p4T10+p6T12)2 |
| 97 | (1+10T+50T2−1078T3−9153T4+45804T5+1496732T6+45804pT7−9153p2T8−1078p3T9+50p4T10+10p5T11+p6T12)2 |
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L(s)=p∏ j=1∏24(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−5.80193236186153991877192141527, −5.64597234276793293522754012365, −5.57950822027377763077666186519, −5.43689528568042674304442276403, −5.35152973487716363715185917574, −5.12045987168770241831574206311, −5.00239101024269964312071468942, −4.86935030706652868948885848305, −4.53011808042202940201052835960, −4.42259064662844709064730415666, −4.35002774704167954760368500884, −4.22186681257837288851574594309, −4.10845597245463350886600356084, −4.10820850581918588614915726140, −3.96505536266266673223687962790, −3.38477227989812797661300606691, −3.37130257409424508583511936349, −3.23476554326180866955618992189, −2.99794960363558667981740078565, −2.62726573415258421154532584130, −2.45183576347053094532963864216, −2.42227100475349896467561560502, −2.35091786896788376344780558035, −1.87502238686416301347343536366, −1.63912336428237542663196072666,
1.63912336428237542663196072666, 1.87502238686416301347343536366, 2.35091786896788376344780558035, 2.42227100475349896467561560502, 2.45183576347053094532963864216, 2.62726573415258421154532584130, 2.99794960363558667981740078565, 3.23476554326180866955618992189, 3.37130257409424508583511936349, 3.38477227989812797661300606691, 3.96505536266266673223687962790, 4.10820850581918588614915726140, 4.10845597245463350886600356084, 4.22186681257837288851574594309, 4.35002774704167954760368500884, 4.42259064662844709064730415666, 4.53011808042202940201052835960, 4.86935030706652868948885848305, 5.00239101024269964312071468942, 5.12045987168770241831574206311, 5.35152973487716363715185917574, 5.43689528568042674304442276403, 5.57950822027377763077666186519, 5.64597234276793293522754012365, 5.80193236186153991877192141527
Plot not available for L-functions of degree greater than 10.