| L(s) = 1 | + 2-s − 3-s − 6-s − 4·7-s − 8-s − 11-s − 8·13-s − 4·14-s − 16-s − 3·17-s + 19-s + 4·21-s − 22-s + 3·23-s + 24-s + 5·25-s − 8·26-s + 27-s − 18·29-s − 2·31-s + 33-s − 3·34-s + 7·37-s + 38-s + 8·39-s − 12·41-s + 4·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 1.51·7-s − 0.353·8-s − 0.301·11-s − 2.21·13-s − 1.06·14-s − 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.872·21-s − 0.213·22-s + 0.625·23-s + 0.204·24-s + 25-s − 1.56·26-s + 0.192·27-s − 3.34·29-s − 0.359·31-s + 0.174·33-s − 0.514·34-s + 1.15·37-s + 0.162·38-s + 1.28·39-s − 1.87·41-s + 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6713388802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6713388802\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.67312866896740766943066607040, −10.84288988036191239679969581820, −10.51363459901816659797583573193, −9.760086933343852107114036719694, −9.606320892641826848155413594912, −9.092519478529019722256134575738, −8.934634046754496247601581557331, −7.74792148853239760290927800229, −7.56197122561348659510900410278, −6.92772657536207404805608061791, −6.75260617019108600773242741599, −5.82573420156529238865791839443, −5.75351273109874385736137096373, −4.98890793324664761846083101511, −4.74805101031080637312037232329, −3.86508632289546169071199358253, −3.47636017692465903494795314282, −2.59051146885381132687769233625, −2.28431823360096414853500230929, −0.44328744894148832756782786247,
0.44328744894148832756782786247, 2.28431823360096414853500230929, 2.59051146885381132687769233625, 3.47636017692465903494795314282, 3.86508632289546169071199358253, 4.74805101031080637312037232329, 4.98890793324664761846083101511, 5.75351273109874385736137096373, 5.82573420156529238865791839443, 6.75260617019108600773242741599, 6.92772657536207404805608061791, 7.56197122561348659510900410278, 7.74792148853239760290927800229, 8.934634046754496247601581557331, 9.092519478529019722256134575738, 9.606320892641826848155413594912, 9.760086933343852107114036719694, 10.51363459901816659797583573193, 10.84288988036191239679969581820, 11.67312866896740766943066607040
