| L(s) = 1 | − 2·3-s + 9-s + 8·11-s − 4·17-s − 10·19-s − 2·25-s + 4·27-s − 16·33-s + 12·41-s − 12·43-s − 49-s + 8·51-s + 20·57-s + 14·59-s − 4·67-s − 12·73-s + 4·75-s − 11·81-s − 10·83-s + 8·89-s + 16·97-s + 8·99-s − 8·107-s + 4·113-s + 26·121-s − 24·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 2.41·11-s − 0.970·17-s − 2.29·19-s − 2/5·25-s + 0.769·27-s − 2.78·33-s + 1.87·41-s − 1.82·43-s − 1/7·49-s + 1.12·51-s + 2.64·57-s + 1.82·59-s − 0.488·67-s − 1.40·73-s + 0.461·75-s − 1.22·81-s − 1.09·83-s + 0.847·89-s + 1.62·97-s + 0.804·99-s − 0.773·107-s + 0.376·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−8.056046829545666874703724421529, −7.23825847434985008786092467133, −6.92350012506979572055111258127, −6.45660456938174818200867480598, −6.26685641768808272430261263019, −5.93748531640264652548242255626, −5.30294344081322495998967730745, −4.58077044621379342973063667196, −4.31934901513247275960207087580, −3.97866110572214522332153756236, −3.31521081303494035975598512057, −2.37074517208793522328749795219, −1.82490893418859332873477226891, −1.03422217455093447514352401670, 0,
1.03422217455093447514352401670, 1.82490893418859332873477226891, 2.37074517208793522328749795219, 3.31521081303494035975598512057, 3.97866110572214522332153756236, 4.31934901513247275960207087580, 4.58077044621379342973063667196, 5.30294344081322495998967730745, 5.93748531640264652548242255626, 6.26685641768808272430261263019, 6.45660456938174818200867480598, 6.92350012506979572055111258127, 7.23825847434985008786092467133, 8.056046829545666874703724421529
