| L(s) = 1 | + 3-s + 2·5-s − 2·9-s − 4·11-s + 2·15-s + 4·23-s + 3·25-s − 5·27-s − 2·31-s − 4·33-s − 4·45-s − 5·49-s + 10·53-s − 8·55-s + 12·59-s − 8·67-s + 4·69-s − 8·71-s + 3·75-s + 81-s + 8·89-s − 2·93-s − 8·97-s + 8·99-s + 8·103-s + 4·113-s + 8·115-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 2/3·9-s − 1.20·11-s + 0.516·15-s + 0.834·23-s + 3/5·25-s − 0.962·27-s − 0.359·31-s − 0.696·33-s − 0.596·45-s − 5/7·49-s + 1.37·53-s − 1.07·55-s + 1.56·59-s − 0.977·67-s + 0.481·69-s − 0.949·71-s + 0.346·75-s + 1/9·81-s + 0.847·89-s − 0.207·93-s − 0.812·97-s + 0.804·99-s + 0.788·103-s + 0.376·113-s + 0.746·115-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
| \(L(1)\) |
\(\approx\) |
\(2.364653730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.364653730\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.910485632671441820589814023221, −7.48871424506948248219696408504, −6.94177623199664755127642909222, −6.63692731360874812577109185274, −5.97751819142824078609103921617, −5.61871087165192419566481586051, −5.31810746726017226321107502670, −4.88997213536455333007486226742, −4.27045632222415094324828946532, −3.66016405613196114074150247744, −3.01766967550677580873809636144, −2.76119936070605038142084898849, −2.19604362376908514442640011347, −1.64495632836103292451752687394, −0.59704093185266703515987131460,
0.59704093185266703515987131460, 1.64495632836103292451752687394, 2.19604362376908514442640011347, 2.76119936070605038142084898849, 3.01766967550677580873809636144, 3.66016405613196114074150247744, 4.27045632222415094324828946532, 4.88997213536455333007486226742, 5.31810746726017226321107502670, 5.61871087165192419566481586051, 5.97751819142824078609103921617, 6.63692731360874812577109185274, 6.94177623199664755127642909222, 7.48871424506948248219696408504, 7.910485632671441820589814023221
