| L(s) = 1 | + 2·2-s − 4·3-s − 14·5-s − 8·6-s + 8·7-s + 27·9-s − 28·10-s + 50·13-s + 16·14-s + 56·15-s − 130·17-s + 54·18-s + 108·19-s − 32·21-s − 384·23-s + 125·25-s + 100·26-s − 142·29-s + 112·30-s − 40·31-s − 32·32-s − 260·34-s − 112·35-s − 382·37-s + 216·38-s − 200·39-s + 118·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s − 1.25·5-s − 0.544·6-s + 0.431·7-s + 9-s − 0.885·10-s + 1.06·13-s + 0.305·14-s + 0.963·15-s − 1.85·17-s + 0.707·18-s + 1.30·19-s − 0.332·21-s − 3.48·23-s + 25-s + 0.754·26-s − 0.909·29-s + 0.681·30-s − 0.231·31-s − 0.176·32-s − 1.31·34-s − 0.540·35-s − 1.69·37-s + 0.922·38-s − 0.821·39-s + 0.449·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.619827533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.619827533\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_4$ | \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 + 4 T - 11 T^{2} - 152 T^{3} - 311 T^{4} - 152 p^{3} T^{5} - 11 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + 14 T + 71 T^{2} - 756 T^{3} - 19459 T^{4} - 756 p^{3} T^{5} + 71 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 8 T - 279 T^{2} + 4976 T^{3} + 55889 T^{4} + 4976 p^{3} T^{5} - 279 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 50 T + 303 T^{2} + 94700 T^{3} - 5400691 T^{4} + 94700 p^{3} T^{5} + 303 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 130 T + 11987 T^{2} + 919620 T^{3} + 60658469 T^{4} + 919620 p^{3} T^{5} + 11987 p^{6} T^{6} + 130 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 108 T + 4805 T^{2} + 221832 T^{3} - 56915351 T^{4} + 221832 p^{3} T^{5} + 4805 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 + 142 T - 4225 T^{2} - 4063188 T^{3} - 473929171 T^{4} - 4063188 p^{3} T^{5} - 4225 p^{6} T^{6} + 142 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 40 T - 28191 T^{2} - 2319280 T^{3} + 747066881 T^{4} - 2319280 p^{3} T^{5} - 28191 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 382 T + 95271 T^{2} + 17044076 T^{3} + 1685075069 T^{4} + 17044076 p^{3} T^{5} + 95271 p^{6} T^{6} + 382 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 118 T - 54997 T^{2} + 14622324 T^{3} + 2065014005 T^{4} + 14622324 p^{3} T^{5} - 54997 p^{6} T^{6} - 118 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 220 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 + 520 T + 166577 T^{2} + 32632080 T^{3} - 325842271 T^{4} + 32632080 p^{3} T^{5} + 166577 p^{6} T^{6} + 520 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 238 T - 92233 T^{2} - 57384180 T^{3} + 73937501 T^{4} - 57384180 p^{3} T^{5} - 92233 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 852 T + 520525 T^{2} - 268504392 T^{3} + 121860838009 T^{4} - 268504392 p^{3} T^{5} + 520525 p^{6} T^{6} - 852 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 190 T - 190881 T^{2} - 79393780 T^{3} + 28241542061 T^{4} - 79393780 p^{3} T^{5} - 190881 p^{6} T^{6} + 190 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - 112 T - 345367 T^{2} + 78767136 T^{3} + 114788729105 T^{4} + 78767136 p^{3} T^{5} - 345367 p^{6} T^{6} - 112 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 6 T - 388981 T^{2} + 4667988 T^{3} + 151292213749 T^{4} + 4667988 p^{3} T^{5} - 388981 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 304 T - 400623 T^{2} - 271673248 T^{3} + 114934095905 T^{4} - 271673248 p^{3} T^{5} - 400623 p^{6} T^{6} + 304 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 820 T + 100613 T^{2} - 386362680 T^{3} - 374346603031 T^{4} - 386362680 p^{3} T^{5} + 100613 p^{6} T^{6} + 820 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 202 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 1406 T + 1064163 T^{2} - 212994940 T^{3} - 671761952059 T^{4} - 212994940 p^{3} T^{5} + 1064163 p^{6} T^{6} - 1406 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−8.233260240178119051393747253499, −8.179360630405472159125264644217, −7.87917157047199845506843814883, −7.40101117247873020266847646418, −7.23746235137263869590346109720, −7.14574926456223037870963612286, −6.85347838884928205803773413064, −6.29492929438826540504998434006, −6.12192781101503726033752511210, −6.00025974663896326888419195834, −5.58194116758163967386631544417, −5.32889959484569569917534993126, −5.00096619583251939948419458862, −4.55448926993931750348501639126, −4.34879524568357516876035850078, −4.12180270625787446021630105617, −3.95052694240735367021785503309, −3.59670847871327053571186721961, −3.29442856817975429712191982488, −2.70117718374539547296103628630, −2.10276461719123841712872999242, −1.88328283767639754539684409935, −1.42087892041095995751315897896, −0.53894441070383035170905213786, −0.48103064208051366432734238737,
0.48103064208051366432734238737, 0.53894441070383035170905213786, 1.42087892041095995751315897896, 1.88328283767639754539684409935, 2.10276461719123841712872999242, 2.70117718374539547296103628630, 3.29442856817975429712191982488, 3.59670847871327053571186721961, 3.95052694240735367021785503309, 4.12180270625787446021630105617, 4.34879524568357516876035850078, 4.55448926993931750348501639126, 5.00096619583251939948419458862, 5.32889959484569569917534993126, 5.58194116758163967386631544417, 6.00025974663896326888419195834, 6.12192781101503726033752511210, 6.29492929438826540504998434006, 6.85347838884928205803773413064, 7.14574926456223037870963612286, 7.23746235137263869590346109720, 7.40101117247873020266847646418, 7.87917157047199845506843814883, 8.179360630405472159125264644217, 8.233260240178119051393747253499
