| L(s) = 1 | − 3·9-s + 4·11-s + 6·13-s − 4·16-s + 5·17-s − 6·19-s − 4·23-s + 25-s + 12·37-s + 12·41-s − 7·49-s − 4·53-s + 12·61-s + 10·67-s − 8·71-s + 6·73-s + 12·83-s − 12·99-s + 6·101-s − 20·113-s − 18·117-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 24·143-s + ⋯ |
| L(s) = 1 | − 9-s + 1.20·11-s + 1.66·13-s − 16-s + 1.21·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s + 1.97·37-s + 1.87·41-s − 49-s − 0.549·53-s + 1.53·61-s + 1.22·67-s − 0.949·71-s + 0.702·73-s + 1.31·83-s − 1.20·99-s + 0.597·101-s − 1.88·113-s − 1.66·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.00·143-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100793 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100793 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567854891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567854891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
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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
−9.468950111629755016423506813444, −9.073603599460326017552793401969, −8.583070359422317617533812227360, −8.133966002350868602260636373252, −7.80931005570661987766891089955, −6.89544851873830150613070757013, −6.36182164990200755878476415274, −6.09682725279302373237806784591, −5.67005340156543048769653310955, −4.75510128894388143670750562067, −4.04737747766069898416270086880, −3.76936690151325547958095557342, −2.87403411704898809246675653157, −2.07389399896790187322502585335, −0.976634163486794973041268442705,
0.976634163486794973041268442705, 2.07389399896790187322502585335, 2.87403411704898809246675653157, 3.76936690151325547958095557342, 4.04737747766069898416270086880, 4.75510128894388143670750562067, 5.67005340156543048769653310955, 6.09682725279302373237806784591, 6.36182164990200755878476415274, 6.89544851873830150613070757013, 7.80931005570661987766891089955, 8.133966002350868602260636373252, 8.583070359422317617533812227360, 9.073603599460326017552793401969, 9.468950111629755016423506813444
