| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s − 2·7-s + 8-s − 2·9-s + 10-s − 7·11-s − 2·12-s − 3·13-s + 2·14-s + 2·15-s − 3·16-s − 5·17-s + 2·18-s − 2·19-s − 20-s + 4·21-s + 7·22-s − 23-s − 2·24-s − 2·25-s + 3·26-s + 10·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 2.11·11-s − 0.577·12-s − 0.832·13-s + 0.534·14-s + 0.516·15-s − 3/4·16-s − 1.21·17-s + 0.471·18-s − 0.458·19-s − 0.223·20-s + 0.872·21-s + 1.49·22-s − 0.208·23-s − 0.408·24-s − 2/5·25-s + 0.588·26-s + 1.92·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100082 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100082 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 163 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
| 307 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 75 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 69 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T - 68 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 131 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−14.6067749764, −13.8917911796, −13.4624305857, −13.1918502872, −12.6538082713, −12.1571783842, −11.7474181520, −11.3222630852, −10.9066791495, −10.5319633045, −10.2251819338, −9.72960840254, −9.04423758996, −8.48724149158, −8.16496488544, −7.54574796548, −7.11704940966, −6.57546183979, −6.02818050085, −5.49716201661, −4.90846907024, −4.56162781217, −3.42995919043, −2.67625323739, −2.10709833535, 0, 0,
2.10709833535, 2.67625323739, 3.42995919043, 4.56162781217, 4.90846907024, 5.49716201661, 6.02818050085, 6.57546183979, 7.11704940966, 7.54574796548, 8.16496488544, 8.48724149158, 9.04423758996, 9.72960840254, 10.2251819338, 10.5319633045, 10.9066791495, 11.3222630852, 11.7474181520, 12.1571783842, 12.6538082713, 13.1918502872, 13.4624305857, 13.8917911796, 14.6067749764
