| L(s) = 1 | + 2·2-s + 2·4-s + 4·7-s + 9-s + 2·11-s + 8·14-s − 4·16-s + 2·18-s + 4·22-s − 12·23-s + 25-s + 8·28-s + 12·29-s − 8·32-s + 2·36-s + 12·37-s − 8·43-s + 4·44-s − 24·46-s + 9·49-s + 2·50-s − 8·53-s + 24·58-s + 4·63-s − 8·64-s + 4·67-s + 8·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 2.13·14-s − 16-s + 0.471·18-s + 0.852·22-s − 2.50·23-s + 1/5·25-s + 1.51·28-s + 2.22·29-s − 1.41·32-s + 1/3·36-s + 1.97·37-s − 1.21·43-s + 0.603·44-s − 3.53·46-s + 9/7·49-s + 0.282·50-s − 1.09·53-s + 3.15·58-s + 0.503·63-s − 64-s + 0.488·67-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.575544077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575544077\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
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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
−9.859772092417623537441417022748, −9.326547603634860485135833727700, −8.492363413233227370023600323835, −8.217875801680133513099365375331, −7.81681140426311786912909363481, −7.00737412264033019137398591174, −6.46828979882124835568402765740, −6.04239278787907299150569896364, −5.44207644392834886456190988700, −4.79124251808362988278289760619, −4.28862990164809284247188295945, −4.10973650596639880735100602431, −3.08004302475956490314706032482, −2.30555874067476859349593118359, −1.44978607363769425296824455358,
1.44978607363769425296824455358, 2.30555874067476859349593118359, 3.08004302475956490314706032482, 4.10973650596639880735100602431, 4.28862990164809284247188295945, 4.79124251808362988278289760619, 5.44207644392834886456190988700, 6.04239278787907299150569896364, 6.46828979882124835568402765740, 7.00737412264033019137398591174, 7.81681140426311786912909363481, 8.217875801680133513099365375331, 8.492363413233227370023600323835, 9.326547603634860485135833727700, 9.859772092417623537441417022748
