| L(s) = 1 | − 2-s − 3·3-s − 4-s − 4·5-s + 3·6-s − 5·7-s + 3·8-s + 2·9-s + 4·10-s − 2·11-s + 3·12-s − 13-s + 5·14-s + 12·15-s − 16-s − 2·17-s − 2·18-s − 4·19-s + 4·20-s + 15·21-s + 2·22-s − 23-s − 9·24-s + 4·25-s + 26-s + 6·27-s + 5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.78·5-s + 1.22·6-s − 1.88·7-s + 1.06·8-s + 2/3·9-s + 1.26·10-s − 0.603·11-s + 0.866·12-s − 0.277·13-s + 1.33·14-s + 3.09·15-s − 1/4·16-s − 0.485·17-s − 0.471·18-s − 0.917·19-s + 0.894·20-s + 3.27·21-s + 0.426·22-s − 0.208·23-s − 1.83·24-s + 4/5·25-s + 0.196·26-s + 1.15·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12092 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12092 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3023 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 28 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 75 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T - 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 171 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + T - 119 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−16.7219894758, −16.4540996741, −15.9520664803, −15.8982824885, −14.9883291917, −14.7516974426, −13.5128415489, −13.3281293192, −12.7086497706, −12.2067678140, −11.8332797994, −11.3212831565, −10.7718093853, −10.3523524322, −9.85307382090, −9.06077272443, −8.61700915306, −7.81343036762, −7.40707075987, −6.60470468155, −6.11765138228, −5.40243275090, −4.52627579575, −3.98202734681, −3.01926525138, 0, 0,
3.01926525138, 3.98202734681, 4.52627579575, 5.40243275090, 6.11765138228, 6.60470468155, 7.40707075987, 7.81343036762, 8.61700915306, 9.06077272443, 9.85307382090, 10.3523524322, 10.7718093853, 11.3212831565, 11.8332797994, 12.2067678140, 12.7086497706, 13.3281293192, 13.5128415489, 14.7516974426, 14.9883291917, 15.8982824885, 15.9520664803, 16.4540996741, 16.7219894758
