| L(s) = 1 | − 3·4-s + 4·5-s + 6·9-s + 24·11-s + 4·16-s − 12·19-s − 12·20-s + 2·25-s − 4·29-s + 8·31-s − 18·36-s − 4·41-s − 72·44-s + 24·45-s − 11·49-s + 96·55-s + 8·59-s − 14·61-s − 9·64-s + 16·71-s + 36·76-s − 28·79-s + 16·80-s + 27·81-s − 6·89-s − 48·95-s + 144·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.78·5-s + 2·9-s + 7.23·11-s + 16-s − 2.75·19-s − 2.68·20-s + 2/5·25-s − 0.742·29-s + 1.43·31-s − 3·36-s − 0.624·41-s − 10.8·44-s + 3.57·45-s − 1.57·49-s + 12.9·55-s + 1.04·59-s − 1.79·61-s − 9/8·64-s + 1.89·71-s + 4.12·76-s − 3.15·79-s + 1.78·80-s + 3·81-s − 0.635·89-s − 4.92·95-s + 14.4·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.851340249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.851340249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.958428636738844619955239356839, −8.263940500713275156923381725042, −8.050574041233869949281626929797, −8.013103161537253337598518569705, −7.09367289743131409736127585181, −7.01729054300237932491353457685, −6.74123410997559027340348339404, −6.64617970890918861813327199336, −6.48596715555925421609951759651, −6.22192516838903531164391268284, −6.05919864597650368186255772752, −5.76042371951542005938839939203, −5.33189844135013584119812763553, −4.57651093879000644031405927739, −4.48740609099643483848199833616, −4.40442794404849228821731066920, −4.23702119251192064689376742855, −3.78787027442736491461540629094, −3.62533051785185703990523860286, −3.59711992163194037596227660269, −2.49441936939358577103692945835, −1.86224008991233329012157758872, −1.56805648833348602971446120159, −1.40762383048239847299199665577, −1.20152323786165677562876649206,
1.20152323786165677562876649206, 1.40762383048239847299199665577, 1.56805648833348602971446120159, 1.86224008991233329012157758872, 2.49441936939358577103692945835, 3.59711992163194037596227660269, 3.62533051785185703990523860286, 3.78787027442736491461540629094, 4.23702119251192064689376742855, 4.40442794404849228821731066920, 4.48740609099643483848199833616, 4.57651093879000644031405927739, 5.33189844135013584119812763553, 5.76042371951542005938839939203, 6.05919864597650368186255772752, 6.22192516838903531164391268284, 6.48596715555925421609951759651, 6.64617970890918861813327199336, 6.74123410997559027340348339404, 7.01729054300237932491353457685, 7.09367289743131409736127585181, 8.013103161537253337598518569705, 8.050574041233869949281626929797, 8.263940500713275156923381725042, 8.958428636738844619955239356839
