| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 7-s + 8-s − 3·9-s + 10-s + 5·11-s − 12-s − 7·13-s − 14-s − 15-s − 16-s − 2·17-s + 3·18-s + 2·19-s + 20-s + 21-s − 5·22-s + 6·23-s + 24-s − 2·25-s + 7·26-s − 4·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 1.94·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.458·19-s + 0.223·20-s + 0.218·21-s − 1.06·22-s + 1.25·23-s + 0.204·24-s − 2/5·25-s + 1.37·26-s − 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3820859774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3820859774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 1109 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 60 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T - 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 47 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 3 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 90 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 264 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−19.5668973954, −19.3951229570, −18.9902525979, −17.9964816252, −17.5062868341, −17.2577837662, −16.7831657294, −15.9158558267, −15.0456568694, −14.7149635447, −14.1550079654, −13.7697677175, −12.7589448000, −12.0130052529, −11.5874409074, −10.9070009166, −9.66881435554, −9.43797063076, −8.76277326868, −8.19769246356, −7.37895581281, −6.52506111333, −5.14556386773, −4.27546771886, −2.82389950240,
2.82389950240, 4.27546771886, 5.14556386773, 6.52506111333, 7.37895581281, 8.19769246356, 8.76277326868, 9.43797063076, 9.66881435554, 10.9070009166, 11.5874409074, 12.0130052529, 12.7589448000, 13.7697677175, 14.1550079654, 14.7149635447, 15.0456568694, 15.9158558267, 16.7831657294, 17.2577837662, 17.5062868341, 17.9964816252, 18.9902525979, 19.3951229570, 19.5668973954
