| L(s) = 1 | − 2·2-s + 2·4-s + 3·5-s + 7-s − 4·8-s + 2·9-s − 6·10-s − 4·11-s − 2·14-s + 8·16-s − 5·17-s − 4·18-s − 7·19-s + 6·20-s + 8·22-s − 2·23-s − 2·25-s + 2·28-s + 29-s − 3·31-s − 8·32-s + 10·34-s + 3·35-s + 4·36-s + 10·37-s + 14·38-s − 12·40-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.34·5-s + 0.377·7-s − 1.41·8-s + 2/3·9-s − 1.89·10-s − 1.20·11-s − 0.534·14-s + 2·16-s − 1.21·17-s − 0.942·18-s − 1.60·19-s + 1.34·20-s + 1.70·22-s − 0.417·23-s − 2/5·25-s + 0.377·28-s + 0.185·29-s − 0.538·31-s − 1.41·32-s + 1.71·34-s + 0.507·35-s + 2/3·36-s + 1.64·37-s + 2.27·38-s − 1.89·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100109 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100109 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 100109 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 162 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 37 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 98 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 185 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 19 T + 224 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−14.3697706620, −13.5634776395, −13.4749190035, −13.0390737483, −12.4609880623, −12.1329707260, −11.4455265142, −10.8913659129, −10.5536664762, −10.1570477186, −9.79793077874, −9.29711421660, −8.95665667334, −8.38806162951, −8.10472917309, −7.45967005268, −6.87336259636, −6.32739074973, −5.76350972725, −5.53018743939, −4.49971680676, −3.99688955844, −2.63016114170, −2.35829655400, −1.54032898213, 0,
1.54032898213, 2.35829655400, 2.63016114170, 3.99688955844, 4.49971680676, 5.53018743939, 5.76350972725, 6.32739074973, 6.87336259636, 7.45967005268, 8.10472917309, 8.38806162951, 8.95665667334, 9.29711421660, 9.79793077874, 10.1570477186, 10.5536664762, 10.8913659129, 11.4455265142, 12.1329707260, 12.4609880623, 13.0390737483, 13.4749190035, 13.5634776395, 14.3697706620
