| L(s) = 1 | − 3·9-s + 4·17-s − 4·19-s − 4·25-s + 6·41-s + 8·43-s + 4·49-s − 16·59-s + 4·67-s − 2·73-s + 9·81-s + 16·83-s − 12·89-s − 10·97-s + 12·107-s − 34·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 9-s + 0.970·17-s − 0.917·19-s − 4/5·25-s + 0.937·41-s + 1.21·43-s + 4/7·49-s − 2.08·59-s + 0.488·67-s − 0.234·73-s + 81-s + 1.75·83-s − 1.27·89-s − 1.01·97-s + 1.16·107-s − 3.19·113-s − 0.545·121-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 − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.80969052675559674658849574077, −7.50348002922080914301099211794, −6.82096588531160295340893232549, −6.38728238601446323303977603504, −5.90815019937903916251268104929, −5.70093967452316249670673962656, −5.14748089057394383625273640312, −4.62245869586976571746118685581, −4.00219381662282336387092715427, −3.69948601322286334271165578008, −2.91917459370557589198931145184, −2.58767256442731917452415918296, −1.89376397280654085668022817676, −1.03807622769668816576661394988, 0,
1.03807622769668816576661394988, 1.89376397280654085668022817676, 2.58767256442731917452415918296, 2.91917459370557589198931145184, 3.69948601322286334271165578008, 4.00219381662282336387092715427, 4.62245869586976571746118685581, 5.14748089057394383625273640312, 5.70093967452316249670673962656, 5.90815019937903916251268104929, 6.38728238601446323303977603504, 6.82096588531160295340893232549, 7.50348002922080914301099211794, 7.80969052675559674658849574077
