| L(s) = 1 | − 4-s − 3·9-s − 4·11-s + 16-s − 12·19-s − 18·29-s + 8·31-s + 3·36-s + 14·41-s + 4·44-s + 20·59-s − 2·61-s − 64-s + 4·71-s + 12·76-s − 20·79-s + 2·89-s + 12·99-s − 6·101-s + 18·109-s + 18·116-s − 10·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 9-s − 1.20·11-s + 1/4·16-s − 2.75·19-s − 3.34·29-s + 1.43·31-s + 1/2·36-s + 2.18·41-s + 0.603·44-s + 2.60·59-s − 0.256·61-s − 1/8·64-s + 0.474·71-s + 1.37·76-s − 2.25·79-s + 0.211·89-s + 1.20·99-s − 0.597·101-s + 1.72·109-s + 1.67·116-s − 0.909·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.813836587196226226741355963366, −8.375993269772858270578799366691, −8.141716446630987438498307722048, −7.51709438543367188847806807998, −7.46695195640450020994060669933, −6.82512007805533580885341957655, −6.27128022298553874526164451131, −5.94132964106281671931828193920, −5.72069370648719354687904553888, −5.20603240951320501985454212796, −4.85335017839183274127865422443, −4.19629552622464142466973928975, −4.00669417617934749328539377673, −3.55541401028347523403780501389, −2.76768342325529352061347305153, −2.31871374831885217339367391228, −2.21241915725933517271838017580, −1.16717194110638346518059984721, 0, 0,
1.16717194110638346518059984721, 2.21241915725933517271838017580, 2.31871374831885217339367391228, 2.76768342325529352061347305153, 3.55541401028347523403780501389, 4.00669417617934749328539377673, 4.19629552622464142466973928975, 4.85335017839183274127865422443, 5.20603240951320501985454212796, 5.72069370648719354687904553888, 5.94132964106281671931828193920, 6.27128022298553874526164451131, 6.82512007805533580885341957655, 7.46695195640450020994060669933, 7.51709438543367188847806807998, 8.141716446630987438498307722048, 8.375993269772858270578799366691, 8.813836587196226226741355963366
