| L(s) = 1 | − 2·3-s − 4-s + 6·7-s + 9-s + 2·12-s + 2·13-s + 16-s − 12·21-s + 25-s + 4·27-s − 6·28-s − 36-s − 10·37-s − 4·39-s − 12·43-s − 2·48-s + 14·49-s − 2·52-s − 4·61-s + 6·63-s − 64-s − 6·67-s + 22·73-s − 2·75-s − 11·81-s + 12·84-s + 12·91-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 2.26·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s + 1/4·16-s − 2.61·21-s + 1/5·25-s + 0.769·27-s − 1.13·28-s − 1/6·36-s − 1.64·37-s − 0.640·39-s − 1.82·43-s − 0.288·48-s + 2·49-s − 0.277·52-s − 0.512·61-s + 0.755·63-s − 1/8·64-s − 0.733·67-s + 2.57·73-s − 0.230·75-s − 1.22·81-s + 1.30·84-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.094516989884481716872748546888, −7.70158203757975610770315706406, −6.93218152688057401872567142051, −6.68770055729895198216161364580, −6.14970985396899297248183326190, −5.37148186282783477747388186443, −5.29117441073688713850944236676, −4.97630027277575908068295367977, −4.44514379194632604451878223603, −3.94426177116844605090509920301, −3.30908920407833803053253029557, −2.42989353523414900304035353582, −1.53347833820744832754504858839, −1.28435873278236245967725570035, 0,
1.28435873278236245967725570035, 1.53347833820744832754504858839, 2.42989353523414900304035353582, 3.30908920407833803053253029557, 3.94426177116844605090509920301, 4.44514379194632604451878223603, 4.97630027277575908068295367977, 5.29117441073688713850944236676, 5.37148186282783477747388186443, 6.14970985396899297248183326190, 6.68770055729895198216161364580, 6.93218152688057401872567142051, 7.70158203757975610770315706406, 8.094516989884481716872748546888
