L(s) = 1 | + 2·4-s + 4·5-s + 2·9-s + 16·11-s + 16-s + 8·19-s + 8·20-s + 8·25-s − 8·29-s − 24·31-s + 4·36-s + 32·44-s + 8·45-s + 20·49-s + 64·55-s + 8·59-s + 12·61-s − 2·64-s + 24·71-s + 16·76-s + 28·79-s + 4·80-s + 81-s − 56·89-s + 32·95-s + 32·99-s + 16·100-s + ⋯ |
L(s) = 1 | + 4-s + 1.78·5-s + 2/3·9-s + 4.82·11-s + 1/4·16-s + 1.83·19-s + 1.78·20-s + 8/5·25-s − 1.48·29-s − 4.31·31-s + 2/3·36-s + 4.82·44-s + 1.19·45-s + 20/7·49-s + 8.62·55-s + 1.04·59-s + 1.53·61-s − 1/4·64-s + 2.84·71-s + 1.83·76-s + 3.15·79-s + 0.447·80-s + 1/9·81-s − 5.93·89-s + 3.28·95-s + 3.21·99-s + 8/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(18.19538666\) |
\(L(\frac12)\) |
\(\approx\) |
\(18.19538666\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 - 4 T + 8 T^{2} + 8 T^{3} - 41 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | \( ( 1 - 2 T + p T^{2} )^{4} \) |
good | 7 | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 13 | \( 1 + 38 T^{2} + 769 T^{4} + 12806 T^{6} + 183028 T^{8} + 12806 p^{2} T^{10} + 769 p^{4} T^{12} + 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 10 T^{2} - 189 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 40 T^{2} + 1071 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 8 T^{2} - 200 T^{3} - 1241 T^{4} - 200 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 38 T^{2} + 1923 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 76 T^{2} + 4095 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 142 T^{2} + 11641 T^{4} + 685150 T^{6} + 31939492 T^{8} + 685150 p^{2} T^{10} + 11641 p^{4} T^{12} + 142 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 104 T^{2} + 4558 T^{4} + 191360 T^{6} + 10390339 T^{8} + 191360 p^{2} T^{10} + 4558 p^{4} T^{12} + 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 36 T^{2} + 1498 T^{4} - 209520 T^{6} - 11490141 T^{8} - 209520 p^{2} T^{10} + 1498 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 4 T - 10 T^{2} + 368 T^{3} - 3749 T^{4} + 368 p T^{5} - 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 6 T - 41 T^{2} + 270 T^{3} + 12 T^{4} + 270 p T^{5} - 41 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 254 T^{2} + 39433 T^{4} + 4090670 T^{6} + 319393444 T^{8} + 4090670 p^{2} T^{10} + 39433 p^{4} T^{12} + 254 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 12 T - 28 T^{2} - 360 T^{3} + 14319 T^{4} - 360 p T^{5} - 28 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 2 p T^{2} + 10033 T^{4} + 1250 p T^{6} - 14685116 T^{8} + 1250 p^{3} T^{10} + 10033 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 14 T + 85 T^{2} + 658 T^{3} - 9404 T^{4} + 658 p T^{5} + 85 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 152 T^{2} + 11778 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 28 T + 416 T^{2} + 5320 T^{3} + 57727 T^{4} + 5320 p T^{5} + 416 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 76 T^{2} - 662 T^{4} + 940880 T^{6} - 101427821 T^{8} + 940880 p^{2} T^{10} - 662 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.82392331952218219030606050288, −4.28231677495373254959296845418, −4.22836728562428842150314098739, −4.20140799789345866755404427896, −4.14612999618081273438881165490, −3.99783345304522377314605243596, −3.81582510883488983519931164169, −3.73134574320151464956804465359, −3.63420317726536867041668183230, −3.58112059221371697357359397729, −3.28370245451862360023301902396, −3.16515501129930089214983614345, −2.79481406502165989973807432770, −2.73389084361539852638982537517, −2.56330472061208517004571420226, −2.33821348194084421293385029238, −2.16568300575501707031925681257, −1.92281266071465925913366163613, −1.91163064807301219454627926060, −1.56037044057019675620847923533, −1.42250200333678320591460541703, −1.33252892601386570058668800920, −1.25678251670961016671837989348, −0.953929358926817826036942558078, −0.47116116087369537822635055725,
0.47116116087369537822635055725, 0.953929358926817826036942558078, 1.25678251670961016671837989348, 1.33252892601386570058668800920, 1.42250200333678320591460541703, 1.56037044057019675620847923533, 1.91163064807301219454627926060, 1.92281266071465925913366163613, 2.16568300575501707031925681257, 2.33821348194084421293385029238, 2.56330472061208517004571420226, 2.73389084361539852638982537517, 2.79481406502165989973807432770, 3.16515501129930089214983614345, 3.28370245451862360023301902396, 3.58112059221371697357359397729, 3.63420317726536867041668183230, 3.73134574320151464956804465359, 3.81582510883488983519931164169, 3.99783345304522377314605243596, 4.14612999618081273438881165490, 4.20140799789345866755404427896, 4.22836728562428842150314098739, 4.28231677495373254959296845418, 4.82392331952218219030606050288
Plot not available for L-functions of degree greater than 10.