| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s + 3·11-s + 2·13-s − 6·17-s − 2·19-s − 6·21-s − 6·23-s + 5·25-s − 9·27-s + 6·29-s − 4·31-s − 9·33-s + 8·37-s − 6·39-s − 9·41-s + 43-s − 6·47-s + 7·49-s + 18·51-s − 24·53-s + 6·57-s − 3·59-s + 8·61-s + 12·63-s − 5·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s + 0.904·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s − 1.30·21-s − 1.25·23-s + 25-s − 1.73·27-s + 1.11·29-s − 0.718·31-s − 1.56·33-s + 1.31·37-s − 0.960·39-s − 1.40·41-s + 0.152·43-s − 0.875·47-s + 49-s + 2.52·51-s − 3.29·53-s + 0.794·57-s − 0.390·59-s + 1.02·61-s + 1.51·63-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9804744772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9804744772\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.08441318358576406178598130481, −10.84765896807616910356500944427, −10.12583989942492981187635496446, −9.763876676698285593475005277032, −9.342056931783938565443015015832, −8.532532932675923376215736275720, −8.486093944904121466651513736447, −7.79570919160403227733370295604, −7.21599666919465477516588905917, −6.57285375393722334313682020533, −6.33817614280598642113746083621, −6.22201153478333084249243393038, −5.34085961246141334135075659352, −4.81852302460133989798006694550, −4.60426906225470960398222912636, −4.02050874707423325095045891584, −3.37227467801038634438479297725, −2.16933233934245689267360742257, −1.60533341219401324620373542745, −0.65765915851346967398994091203,
0.65765915851346967398994091203, 1.60533341219401324620373542745, 2.16933233934245689267360742257, 3.37227467801038634438479297725, 4.02050874707423325095045891584, 4.60426906225470960398222912636, 4.81852302460133989798006694550, 5.34085961246141334135075659352, 6.22201153478333084249243393038, 6.33817614280598642113746083621, 6.57285375393722334313682020533, 7.21599666919465477516588905917, 7.79570919160403227733370295604, 8.486093944904121466651513736447, 8.532532932675923376215736275720, 9.342056931783938565443015015832, 9.763876676698285593475005277032, 10.12583989942492981187635496446, 10.84765896807616910356500944427, 11.08441318358576406178598130481
