| L(s) = 1 | − 4·5-s + 9-s + 4·13-s − 8·17-s + 11·25-s − 4·29-s + 8·37-s + 4·41-s − 4·45-s − 49-s − 8·53-s − 20·61-s − 16·65-s + 24·73-s + 81-s + 32·85-s + 12·89-s + 16·97-s + 4·101-s − 12·109-s − 16·113-s + 4·117-s − 6·121-s − 24·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s + 1.10·13-s − 1.94·17-s + 11/5·25-s − 0.742·29-s + 1.31·37-s + 0.624·41-s − 0.596·45-s − 1/7·49-s − 1.09·53-s − 2.56·61-s − 1.98·65-s + 2.80·73-s + 1/9·81-s + 3.47·85-s + 1.27·89-s + 1.62·97-s + 0.398·101-s − 1.14·109-s − 1.50·113-s + 0.369·117-s − 0.545·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.012912829647167215704760004708, −7.71449162947661836420265068332, −7.33372368484635249387450536854, −6.75823637960211716328340425718, −6.28941019972383857037754310409, −6.10816158932353051957037945885, −5.10302467505602603374268699583, −4.70725572561342107049587651940, −4.26573810362333541058142475683, −3.85072701399017100200207637759, −3.42354699340071091780307994487, −2.73359461063529602840073663977, −1.97106312851700076725021669276, −0.999306442120385292060302743434, 0,
0.999306442120385292060302743434, 1.97106312851700076725021669276, 2.73359461063529602840073663977, 3.42354699340071091780307994487, 3.85072701399017100200207637759, 4.26573810362333541058142475683, 4.70725572561342107049587651940, 5.10302467505602603374268699583, 6.10816158932353051957037945885, 6.28941019972383857037754310409, 6.75823637960211716328340425718, 7.33372368484635249387450536854, 7.71449162947661836420265068332, 8.012912829647167215704760004708
