| L(s) = 1 | − 2·3-s + 5-s − 4·7-s + 3·9-s + 11-s + 5·13-s − 2·15-s − 2·17-s − 7·19-s + 8·21-s + 3·23-s − 25-s − 4·27-s − 2·29-s + 4·31-s − 2·33-s − 4·35-s + 8·37-s − 10·39-s + 13·41-s − 5·43-s + 3·45-s + 8·47-s − 2·49-s + 4·51-s + 2·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.51·7-s + 9-s + 0.301·11-s + 1.38·13-s − 0.516·15-s − 0.485·17-s − 1.60·19-s + 1.74·21-s + 0.625·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s − 0.676·35-s + 1.31·37-s − 1.60·39-s + 2.03·41-s − 0.762·43-s + 0.447·45-s + 1.16·47-s − 2/7·49-s + 0.560·51-s + 0.274·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.260969315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260969315\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 13 T + 116 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 158 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.423839731201200949787747354869, −9.258270811267665938832426139767, −9.076766689812169277984020130586, −8.455771514851122540477265146428, −7.83206741167359865274320277734, −7.70794491855369412083969717955, −6.84489037128877383118142557845, −6.49770179300622676206087142467, −6.34243639038922239188155970260, −6.20215982376736692113273062931, −5.65422544209652202219381272651, −5.18594745267568855722830012992, −4.39581212803672445485453074910, −4.38937750389360942099431779791, −3.51141735467351448762260189680, −3.44215440251122875846054022261, −2.41138834711447015093842910952, −2.11239890130400892477352443269, −1.08190774970485546909395775690, −0.55094650928096313460570718003,
0.55094650928096313460570718003, 1.08190774970485546909395775690, 2.11239890130400892477352443269, 2.41138834711447015093842910952, 3.44215440251122875846054022261, 3.51141735467351448762260189680, 4.38937750389360942099431779791, 4.39581212803672445485453074910, 5.18594745267568855722830012992, 5.65422544209652202219381272651, 6.20215982376736692113273062931, 6.34243639038922239188155970260, 6.49770179300622676206087142467, 6.84489037128877383118142557845, 7.70794491855369412083969717955, 7.83206741167359865274320277734, 8.455771514851122540477265146428, 9.076766689812169277984020130586, 9.258270811267665938832426139767, 9.423839731201200949787747354869
