| L(s) = 1 | − 2·3-s + 3·5-s + 2·7-s + 3·9-s + 2·11-s + 5·13-s − 6·15-s + 4·17-s − 19-s − 4·21-s + 25-s − 4·27-s + 5·29-s − 2·31-s − 4·33-s + 6·35-s − 37-s − 10·39-s + 6·41-s − 4·43-s + 9·45-s − 5·47-s + 3·49-s − 8·51-s + 14·53-s + 6·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s + 0.755·7-s + 9-s + 0.603·11-s + 1.38·13-s − 1.54·15-s + 0.970·17-s − 0.229·19-s − 0.872·21-s + 1/5·25-s − 0.769·27-s + 0.928·29-s − 0.359·31-s − 0.696·33-s + 1.01·35-s − 0.164·37-s − 1.60·39-s + 0.937·41-s − 0.609·43-s + 1.34·45-s − 0.729·47-s + 3/7·49-s − 1.12·51-s + 1.92·53-s + 0.809·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.249945971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.249945971\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 130 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 11 T + 172 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−9.518589989847936184268152410446, −9.116193576874409977443175391910, −8.549375854566871081684669441821, −8.507832951465863837984304384947, −7.64765863653229289799738742963, −7.62741994722189564411576355492, −6.82956731029780470138115134481, −6.55597401311922205622931187793, −6.08903391970779655645997043621, −5.96964190422042817151361863277, −5.33783062418990559981074772155, −5.27716616218414770588336266537, −4.60091343536064424272029901453, −4.19370202042418280146969324201, −3.60205162629982707605815095299, −3.22135372980770518582582174202, −2.08177048337354663296045191519, −2.04204062564621713530009498999, −1.13595700592632520635733462762, −0.884046434616091947679349570291,
0.884046434616091947679349570291, 1.13595700592632520635733462762, 2.04204062564621713530009498999, 2.08177048337354663296045191519, 3.22135372980770518582582174202, 3.60205162629982707605815095299, 4.19370202042418280146969324201, 4.60091343536064424272029901453, 5.27716616218414770588336266537, 5.33783062418990559981074772155, 5.96964190422042817151361863277, 6.08903391970779655645997043621, 6.55597401311922205622931187793, 6.82956731029780470138115134481, 7.62741994722189564411576355492, 7.64765863653229289799738742963, 8.507832951465863837984304384947, 8.549375854566871081684669441821, 9.116193576874409977443175391910, 9.518589989847936184268152410446
