| L(s) = 1 | − 3-s + 3·5-s − 2·9-s − 2·11-s − 3·15-s − 4·23-s + 4·25-s + 5·27-s − 5·31-s + 2·33-s + 6·37-s − 6·45-s − 6·47-s + 10·49-s + 5·53-s − 6·55-s − 6·59-s + 7·67-s + 4·69-s − 7·71-s − 4·75-s + 81-s + 13·89-s + 5·93-s + 28·97-s + 4·99-s − 7·103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s − 0.603·11-s − 0.774·15-s − 0.834·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s + 0.348·33-s + 0.986·37-s − 0.894·45-s − 0.875·47-s + 10/7·49-s + 0.686·53-s − 0.809·55-s − 0.781·59-s + 0.855·67-s + 0.481·69-s − 0.830·71-s − 0.461·75-s + 1/9·81-s + 1.37·89-s + 0.518·93-s + 2.84·97-s + 0.402·99-s − 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
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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
−7.51515793913511756630524359722, −7.24552142899280262554623403459, −6.49927934915805437801475102058, −6.22643770966893595214823620399, −5.94411523242652267267775321418, −5.48701117724538981355717252548, −5.10674837259636542814978370388, −4.78113628716589885422144688761, −3.99120132485445483910941867880, −3.56024692584848914413275474713, −2.71721447307338068828422026441, −2.45489459914752010840607708069, −1.84610986203243854483088274610, −1.04691632933745722708988300066, 0,
1.04691632933745722708988300066, 1.84610986203243854483088274610, 2.45489459914752010840607708069, 2.71721447307338068828422026441, 3.56024692584848914413275474713, 3.99120132485445483910941867880, 4.78113628716589885422144688761, 5.10674837259636542814978370388, 5.48701117724538981355717252548, 5.94411523242652267267775321418, 6.22643770966893595214823620399, 6.49927934915805437801475102058, 7.24552142899280262554623403459, 7.51515793913511756630524359722
