| L(s) = 1 | − 2-s − 3·3-s + 4-s − 3·5-s + 3·6-s − 5·7-s + 8-s + 3·9-s + 3·10-s + 11-s − 3·12-s − 13-s + 5·14-s + 9·15-s − 3·16-s − 3·18-s − 3·20-s + 15·21-s − 22-s − 3·24-s + 2·25-s + 26-s − 5·28-s + 5·29-s − 9·30-s − 4·31-s + 5·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 1.88·7-s + 0.353·8-s + 9-s + 0.948·10-s + 0.301·11-s − 0.866·12-s − 0.277·13-s + 1.33·14-s + 2.32·15-s − 3/4·16-s − 0.707·18-s − 0.670·20-s + 3.27·21-s − 0.213·22-s − 0.612·24-s + 2/5·25-s + 0.196·26-s − 0.944·28-s + 0.928·29-s − 1.64·30-s − 0.718·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2218 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2218 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 1109 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 16 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 79 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 50 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−19.0605488331, −18.3219294062, −17.9420453395, −17.1164408142, −16.7847846659, −16.4497548636, −15.9629836182, −15.6246660168, −14.9929509700, −13.9671208256, −13.2166322025, −12.5574726842, −12.1086231199, −11.5465556112, −11.2014666482, −10.4513024713, −9.95982966139, −9.24051128999, −8.33233825697, −7.49584682237, −6.74252698138, −6.41507398795, −5.44412805117, −4.37193481560, −3.28386456588, 0,
3.28386456588, 4.37193481560, 5.44412805117, 6.41507398795, 6.74252698138, 7.49584682237, 8.33233825697, 9.24051128999, 9.95982966139, 10.4513024713, 11.2014666482, 11.5465556112, 12.1086231199, 12.5574726842, 13.2166322025, 13.9671208256, 14.9929509700, 15.6246660168, 15.9629836182, 16.4497548636, 16.7847846659, 17.1164408142, 17.9420453395, 18.3219294062, 19.0605488331
