| L(s) = 1 | + 2-s + 4-s + 5·5-s + 8-s + 9-s + 5·10-s + 5·11-s + 16-s − 3·17-s + 18-s + 5·20-s + 5·22-s + 9·25-s + 5·29-s + 32-s − 3·34-s + 36-s − 5·37-s + 5·40-s + 5·44-s + 5·45-s + 47-s − 5·49-s + 9·50-s + 25·55-s + 5·58-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 2.23·5-s + 0.353·8-s + 1/3·9-s + 1.58·10-s + 1.50·11-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.11·20-s + 1.06·22-s + 9/5·25-s + 0.928·29-s + 0.176·32-s − 0.514·34-s + 1/6·36-s − 0.821·37-s + 0.790·40-s + 0.753·44-s + 0.745·45-s + 0.145·47-s − 5/7·49-s + 1.27·50-s + 3.37·55-s + 0.656·58-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.987277977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.987277977\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
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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
−8.975581314241574786594883080899, −8.370020113224032896501672263580, −7.82399744855883057514534251127, −6.96559002430934576277437530054, −6.75160134147257771622726175272, −6.30214699341386463679086792862, −6.03922367518143702027240132897, −5.37109918346134315245213618504, −5.08756156915922341035296094669, −4.26459618706818661825630979252, −3.97929750865036667897427199083, −3.03609892798910970439553353760, −2.48355061711640472262958504374, −1.68866933584723239579667054942, −1.43368979612908446377616013521,
1.43368979612908446377616013521, 1.68866933584723239579667054942, 2.48355061711640472262958504374, 3.03609892798910970439553353760, 3.97929750865036667897427199083, 4.26459618706818661825630979252, 5.08756156915922341035296094669, 5.37109918346134315245213618504, 6.03922367518143702027240132897, 6.30214699341386463679086792862, 6.75160134147257771622726175272, 6.96559002430934576277437530054, 7.82399744855883057514534251127, 8.370020113224032896501672263580, 8.975581314241574786594883080899
