| L(s) = 1 | − 2-s − 4·3-s − 6·5-s + 4·6-s − 3·7-s + 8-s + 7·9-s + 6·10-s − 4·11-s + 2·13-s + 3·14-s + 24·15-s − 16-s − 17-s − 7·18-s − 6·19-s + 12·21-s + 4·22-s − 23-s − 4·24-s + 18·25-s − 2·26-s − 4·27-s − 6·29-s − 24·30-s − 2·31-s + 16·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 2.68·5-s + 1.63·6-s − 1.13·7-s + 0.353·8-s + 7/3·9-s + 1.89·10-s − 1.20·11-s + 0.554·13-s + 0.801·14-s + 6.19·15-s − 1/4·16-s − 0.242·17-s − 1.64·18-s − 1.37·19-s + 2.61·21-s + 0.852·22-s − 0.208·23-s − 0.816·24-s + 18/5·25-s − 0.392·26-s − 0.769·27-s − 1.11·29-s − 4.38·30-s − 0.359·31-s + 2.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8452 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8452 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 2113 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 34 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 146 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−17.2236475593, −16.5797390066, −16.3888370194, −16.1117264558, −15.6038907382, −15.2331205320, −14.6982031675, −13.3882860932, −12.9749924860, −12.5004054022, −11.9829012467, −11.6492727686, −11.0046335153, −10.8317857836, −10.4400187517, −9.51018460387, −8.68842897060, −8.02464277038, −7.66811348541, −6.85547926913, −6.37442807639, −5.61314679120, −4.77809428916, −4.14393091775, −3.31956371415, 0, 0,
3.31956371415, 4.14393091775, 4.77809428916, 5.61314679120, 6.37442807639, 6.85547926913, 7.66811348541, 8.02464277038, 8.68842897060, 9.51018460387, 10.4400187517, 10.8317857836, 11.0046335153, 11.6492727686, 11.9829012467, 12.5004054022, 12.9749924860, 13.3882860932, 14.6982031675, 15.2331205320, 15.6038907382, 16.1117264558, 16.3888370194, 16.5797390066, 17.2236475593
