| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 13-s + 16-s + 9·17-s − 3·20-s − 25-s − 26-s + 3·29-s − 32-s − 9·34-s + 4·37-s + 3·40-s + 6·41-s + 5·49-s + 50-s + 52-s + 9·53-s − 3·58-s + 7·61-s + 64-s − 3·65-s + 9·68-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 0.277·13-s + 1/4·16-s + 2.18·17-s − 0.670·20-s − 1/5·25-s − 0.196·26-s + 0.557·29-s − 0.176·32-s − 1.54·34-s + 0.657·37-s + 0.474·40-s + 0.937·41-s + 5/7·49-s + 0.141·50-s + 0.138·52-s + 1.23·53-s − 0.393·58-s + 0.896·61-s + 1/8·64-s − 0.372·65-s + 1.09·68-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6994234296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6994234296\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.71078943898419379646698133464, −10.12246553244111829148511353101, −9.774343537705347867383761445616, −9.099009285845851190925400695426, −8.398892770471607611995591587847, −8.075563580932278558882011707087, −7.43987667222868071998828170578, −7.31030327344638267995054576950, −6.25458456523803888370903163061, −5.74518430107797086806283697356, −4.95388126329155770359228838926, −3.93138368104361941401452234232, −3.56647172461029401065873888149, −2.55537425144836814611645749087, −1.00052051447345267581163311920,
1.00052051447345267581163311920, 2.55537425144836814611645749087, 3.56647172461029401065873888149, 3.93138368104361941401452234232, 4.95388126329155770359228838926, 5.74518430107797086806283697356, 6.25458456523803888370903163061, 7.31030327344638267995054576950, 7.43987667222868071998828170578, 8.075563580932278558882011707087, 8.398892770471607611995591587847, 9.099009285845851190925400695426, 9.774343537705347867383761445616, 10.12246553244111829148511353101, 10.71078943898419379646698133464
