| L(s) = 1 | + 10·7-s − 6·9-s − 10·13-s − 8·16-s − 2·19-s − 14·31-s + 20·37-s + 50·49-s − 60·63-s − 10·67-s + 20·73-s + 27·81-s − 100·91-s + 10·97-s − 38·109-s − 80·112-s + 60·117-s + 127-s + 131-s − 20·133-s + 137-s + 139-s + 48·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3.77·7-s − 2·9-s − 2.77·13-s − 2·16-s − 0.458·19-s − 2.51·31-s + 3.28·37-s + 50/7·49-s − 7.55·63-s − 1.22·67-s + 2.34·73-s + 3·81-s − 10.4·91-s + 1.01·97-s − 3.63·109-s − 7.55·112-s + 5.54·117-s + 0.0887·127-s + 0.0873·131-s − 1.73·133-s + 0.0854·137-s + 0.0848·139-s + 4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.490771676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.490771676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 - 109 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 - 169 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.27328914538756912438143028987, −7.05770988388194383804801191386, −6.82197504579202096549246691743, −6.58255775500668703481249970398, −6.11319094713825344935056219773, −6.03841540027783132299725938131, −5.54417976903032211066140847410, −5.48164636197832390824368250052, −5.35891493113113922688340785406, −5.11544503807145638139873431174, −4.84364880993039167376461831771, −4.50483074601416344641948005388, −4.47434675028010134372696776975, −4.31469501421071280506919229548, −4.21293396735874632103657674291, −3.42802555286792460338528000976, −3.30439494823256701558797315501, −2.74064560888535312860770154905, −2.64208281797026323036628464654, −2.18812974138408261539050383892, −2.12119300948260260160194719861, −1.90536212359161802542157471214, −1.64940822734384121253973934642, −0.74105292342560875808386332519, −0.44082746659005415644690551805,
0.44082746659005415644690551805, 0.74105292342560875808386332519, 1.64940822734384121253973934642, 1.90536212359161802542157471214, 2.12119300948260260160194719861, 2.18812974138408261539050383892, 2.64208281797026323036628464654, 2.74064560888535312860770154905, 3.30439494823256701558797315501, 3.42802555286792460338528000976, 4.21293396735874632103657674291, 4.31469501421071280506919229548, 4.47434675028010134372696776975, 4.50483074601416344641948005388, 4.84364880993039167376461831771, 5.11544503807145638139873431174, 5.35891493113113922688340785406, 5.48164636197832390824368250052, 5.54417976903032211066140847410, 6.03841540027783132299725938131, 6.11319094713825344935056219773, 6.58255775500668703481249970398, 6.82197504579202096549246691743, 7.05770988388194383804801191386, 7.27328914538756912438143028987
