L(s) = 1 | + 2-s + 3-s + 5-s + 6-s + 2·7-s + 8-s + 9-s + 10-s − 3·11-s − 3·13-s + 2·14-s + 15-s − 16-s − 17-s + 18-s − 6·19-s + 2·21-s − 3·22-s + 23-s + 24-s − 4·25-s − 3·26-s + 4·27-s + 29-s + 30-s − 31-s − 6·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.832·13-s + 0.534·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.37·19-s + 0.436·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.588·26-s + 0.769·27-s + 0.185·29-s + 0.182·30-s − 0.179·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15065 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15065 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.856595216\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.856595216\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 131 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 21 T + p T^{2} ) \) |
good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 17 T + 248 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−15.7958732973, −15.5581788988, −14.9376334084, −14.6125650362, −14.0236070994, −13.7890876744, −13.2700533048, −12.6786239931, −12.5310630481, −11.6778357290, −11.0905799377, −10.5132108608, −10.2624692610, −9.38367128345, −8.97307985111, −8.27488580249, −7.75265267990, −7.24163743293, −6.48303493982, −5.69768177964, −4.95224823441, −4.55784168394, −3.84903191454, −2.58483783931, −2.03053541052,
2.03053541052, 2.58483783931, 3.84903191454, 4.55784168394, 4.95224823441, 5.69768177964, 6.48303493982, 7.24163743293, 7.75265267990, 8.27488580249, 8.97307985111, 9.38367128345, 10.2624692610, 10.5132108608, 11.0905799377, 11.6778357290, 12.5310630481, 12.6786239931, 13.2700533048, 13.7890876744, 14.0236070994, 14.6125650362, 14.9376334084, 15.5581788988, 15.7958732973