| L(s) = 1 | + 2·5-s + 3·7-s − 3·11-s + 9·17-s − 19-s − 12·23-s − 7·25-s + 2·29-s + 31-s + 6·35-s − 27·37-s + 30·41-s − 15·47-s + 7·49-s − 3·53-s − 6·55-s − 59-s + 12·61-s + 18·67-s − 12·71-s + 15·73-s − 9·77-s + 7·79-s + 18·85-s + 14·89-s − 2·95-s − 12·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.13·7-s − 0.904·11-s + 2.18·17-s − 0.229·19-s − 2.50·23-s − 7/5·25-s + 0.371·29-s + 0.179·31-s + 1.01·35-s − 4.43·37-s + 4.68·41-s − 2.18·47-s + 49-s − 0.412·53-s − 0.809·55-s − 0.130·59-s + 1.53·61-s + 2.19·67-s − 1.42·71-s + 1.75·73-s − 1.02·77-s + 0.787·79-s + 1.95·85-s + 1.48·89-s − 0.205·95-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.890394427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.890394427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 3 T - T^{2} - 36 T^{3} - 120 T^{4} - 36 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 9 T + 63 T^{2} - 324 T^{3} + 1466 T^{4} - 324 p T^{5} + 63 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + T - 23 T^{2} - 14 T^{3} + 196 T^{4} - 14 p T^{5} - 23 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_{4}$ | \( ( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - T - 47 T^{2} + 14 T^{3} + 1312 T^{4} + 14 p T^{5} - 47 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 27 T + 373 T^{2} + 3510 T^{3} + 24522 T^{4} + 3510 p T^{5} + 373 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 15 T + 183 T^{2} + 1620 T^{3} + 12980 T^{4} + 1620 p T^{5} + 183 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 3 T + 67 T^{2} + 192 T^{3} + 1446 T^{4} + 192 p T^{5} + 67 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + T - 103 T^{2} - 14 T^{3} + 7276 T^{4} - 14 p T^{5} - 103 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 43 T^{2} + 252 T^{3} - 2304 T^{4} + 252 p T^{5} + 43 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9972 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 15 T + 235 T^{2} - 2400 T^{3} + 25746 T^{4} - 2400 p T^{5} + 235 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 7 T - 107 T^{2} + 14 T^{3} + 14224 T^{4} + 14 p T^{5} - 107 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.87472863702048112322609985758, −6.53054128630629453331654772679, −6.51873812299085196244697985307, −6.28536566825912426675543054864, −5.90529946929309367687627780349, −5.80753402185368277305551355514, −5.49007115178164561047476023306, −5.33537280797048194820378049673, −5.30863740790176949616251094760, −5.11575367310599332444029556229, −4.62384029694214488069549362008, −4.55027447178937623427004660504, −4.02158649334980947456875221999, −3.91574818970243426886368728808, −3.77134714037811099761921947160, −3.43764124208605952986862173797, −3.25928527063477658983220006936, −2.63643092702922363064317743366, −2.54293494173146121070692595021, −2.22475336630229759753527492644, −1.97673215619286033884015768245, −1.59141598350467229142985266217, −1.51373439891561765317885785346, −0.898795778203968080107793765437, −0.35390503619542472504931198482,
0.35390503619542472504931198482, 0.898795778203968080107793765437, 1.51373439891561765317885785346, 1.59141598350467229142985266217, 1.97673215619286033884015768245, 2.22475336630229759753527492644, 2.54293494173146121070692595021, 2.63643092702922363064317743366, 3.25928527063477658983220006936, 3.43764124208605952986862173797, 3.77134714037811099761921947160, 3.91574818970243426886368728808, 4.02158649334980947456875221999, 4.55027447178937623427004660504, 4.62384029694214488069549362008, 5.11575367310599332444029556229, 5.30863740790176949616251094760, 5.33537280797048194820378049673, 5.49007115178164561047476023306, 5.80753402185368277305551355514, 5.90529946929309367687627780349, 6.28536566825912426675543054864, 6.51873812299085196244697985307, 6.53054128630629453331654772679, 6.87472863702048112322609985758
