| L(s) = 1 | − 2·2-s + 3-s + 4-s − 5-s − 2·6-s + 4·7-s + 2·10-s + 12-s − 7·13-s − 8·14-s − 15-s + 16-s − 8·17-s + 3·19-s − 20-s + 4·21-s + 2·23-s − 2·25-s + 14·26-s + 2·27-s + 4·28-s − 29-s + 2·30-s + 7·31-s + 2·32-s + 16·34-s − 4·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 1.51·7-s + 0.632·10-s + 0.288·12-s − 1.94·13-s − 2.13·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.688·19-s − 0.223·20-s + 0.872·21-s + 0.417·23-s − 2/5·25-s + 2.74·26-s + 0.384·27-s + 0.755·28-s − 0.185·29-s + 0.365·30-s + 1.25·31-s + 0.353·32-s + 2.74·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1069 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1069 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3048376776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3048376776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 1069 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 41 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 7 T + p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 17 T + 173 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
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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
−19.6794700020, −19.3726883943, −18.7713950190, −18.1082163553, −17.5370526833, −17.4929062612, −16.9399225227, −15.8295130131, −15.4556939456, −14.7414444245, −14.3534070546, −13.6812043474, −12.8419182354, −11.8463677138, −11.6094229166, −10.7214648671, −9.97882413241, −9.33355739953, −8.66268856987, −8.24272698354, −7.55791360582, −6.84761399745, −5.13903222896, −4.41422726903, −2.44677381547,
2.44677381547, 4.41422726903, 5.13903222896, 6.84761399745, 7.55791360582, 8.24272698354, 8.66268856987, 9.33355739953, 9.97882413241, 10.7214648671, 11.6094229166, 11.8463677138, 12.8419182354, 13.6812043474, 14.3534070546, 14.7414444245, 15.4556939456, 15.8295130131, 16.9399225227, 17.4929062612, 17.5370526833, 18.1082163553, 18.7713950190, 19.3726883943, 19.6794700020
