| L(s) = 1 | − 4·5-s + 9-s − 16·23-s + 2·25-s + 16·31-s + 12·37-s − 6·41-s + 8·43-s − 4·45-s − 14·49-s + 8·59-s − 4·61-s + 20·73-s + 81-s − 8·83-s + 32·103-s − 24·107-s + 36·113-s + 64·115-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s − 3.33·23-s + 2/5·25-s + 2.87·31-s + 1.97·37-s − 0.937·41-s + 1.21·43-s − 0.596·45-s − 2·49-s + 1.04·59-s − 0.512·61-s + 2.34·73-s + 1/9·81-s − 0.878·83-s + 3.15·103-s − 2.32·107-s + 3.38·113-s + 5.96·115-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 41 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.026186869701343793995719370946, −7.69871195882998409707078390213, −7.22146819849449730317844958445, −6.42897107072744719896577758454, −6.27790065804131757995134714208, −5.87374692001595525036599680786, −5.02735961821708020838141617431, −4.45207306839759870292806796179, −4.25303028692796488061253338187, −3.80282292252027420837460709024, −3.34759720889321445630702897390, −2.53135018330097873522387472392, −1.99438532312917026138235360361, −0.899616754593221603136443851369, 0,
0.899616754593221603136443851369, 1.99438532312917026138235360361, 2.53135018330097873522387472392, 3.34759720889321445630702897390, 3.80282292252027420837460709024, 4.25303028692796488061253338187, 4.45207306839759870292806796179, 5.02735961821708020838141617431, 5.87374692001595525036599680786, 6.27790065804131757995134714208, 6.42897107072744719896577758454, 7.22146819849449730317844958445, 7.69871195882998409707078390213, 8.026186869701343793995719370946
