| L(s) = 1 | − 2·3-s − 2·5-s − 4·7-s + 9-s − 11-s − 6·13-s + 4·15-s + 5·17-s − 6·19-s + 8·21-s + 6·23-s + 4·25-s + 4·27-s + 29-s − 5·31-s + 2·33-s + 8·35-s + 37-s + 12·39-s − 41-s − 9·43-s − 2·45-s − 13·47-s + 6·49-s − 10·51-s + 2·55-s + 12·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 1.03·15-s + 1.21·17-s − 1.37·19-s + 1.74·21-s + 1.25·23-s + 4/5·25-s + 0.769·27-s + 0.185·29-s − 0.898·31-s + 0.348·33-s + 1.35·35-s + 0.164·37-s + 1.92·39-s − 0.156·41-s − 1.37·43-s − 0.298·45-s − 1.89·47-s + 6/7·49-s − 1.40·51-s + 0.269·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 82 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 96 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 19 T + 204 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−17.2023405045, −16.8482493977, −16.6873689184, −16.2591705204, −15.5423915949, −15.0775890188, −14.6681950222, −14.0388478642, −12.9310358989, −12.7247376150, −12.5063319023, −11.7290851083, −11.3609233088, −10.6217950714, −10.1272250252, −9.65513813235, −8.89113133673, −8.07117990284, −7.38490535339, −6.67376499949, −6.36296694387, −5.20184409837, −4.91068971172, −3.65362251723, −2.84588893439, 0,
2.84588893439, 3.65362251723, 4.91068971172, 5.20184409837, 6.36296694387, 6.67376499949, 7.38490535339, 8.07117990284, 8.89113133673, 9.65513813235, 10.1272250252, 10.6217950714, 11.3609233088, 11.7290851083, 12.5063319023, 12.7247376150, 12.9310358989, 14.0388478642, 14.6681950222, 15.0775890188, 15.5423915949, 16.2591705204, 16.6873689184, 16.8482493977, 17.2023405045
