| L(s) = 1 | + 2-s − 3·3-s − 3·6-s − 2·7-s + 8-s + 3·9-s − 5·13-s − 2·14-s − 16-s + 3·18-s + 2·19-s + 6·21-s − 3·23-s − 3·24-s − 7·25-s − 5·26-s + 29-s − 3·31-s − 6·32-s − 2·37-s + 2·38-s + 15·39-s − 3·41-s + 6·42-s + 2·43-s − 3·46-s − 3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1.22·6-s − 0.755·7-s + 0.353·8-s + 9-s − 1.38·13-s − 0.534·14-s − 1/4·16-s + 0.707·18-s + 0.458·19-s + 1.30·21-s − 0.625·23-s − 0.612·24-s − 7/5·25-s − 0.980·26-s + 0.185·29-s − 0.538·31-s − 1.06·32-s − 0.328·37-s + 0.324·38-s + 2.40·39-s − 0.468·41-s + 0.925·42-s + 0.304·43-s − 0.442·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15073 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15073 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 15073 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 98 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - T - 39 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T - T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 90 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 125 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 107 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 82 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−16.3886434250, −15.8662283355, −15.5877560902, −14.8391579207, −14.3140118090, −13.8071572634, −13.3914167383, −12.7940786696, −12.3173879598, −11.9646143890, −11.5220724326, −11.0261308993, −10.4639368300, −9.79727299764, −9.59591874840, −8.69627373787, −7.74311586674, −7.27021574245, −6.59290232292, −6.04662431839, −5.40354856638, −5.06605606303, −4.29897258551, −3.51028666353, −2.20738320792, 0,
2.20738320792, 3.51028666353, 4.29897258551, 5.06605606303, 5.40354856638, 6.04662431839, 6.59290232292, 7.27021574245, 7.74311586674, 8.69627373787, 9.59591874840, 9.79727299764, 10.4639368300, 11.0261308993, 11.5220724326, 11.9646143890, 12.3173879598, 12.7940786696, 13.3914167383, 13.8071572634, 14.3140118090, 14.8391579207, 15.5877560902, 15.8662283355, 16.3886434250
