| L(s) = 1 | + 3·3-s − 3·5-s + 3·7-s + 3·9-s + 6·11-s − 5·13-s − 9·15-s + 2·17-s + 3·19-s + 9·21-s + 23-s − 25-s − 29-s + 8·31-s + 18·33-s − 9·35-s + 6·37-s − 15·39-s + 6·41-s − 2·43-s − 9·45-s + 3·47-s + 5·49-s + 6·51-s − 14·53-s − 18·55-s + 9·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 1.13·7-s + 9-s + 1.80·11-s − 1.38·13-s − 2.32·15-s + 0.485·17-s + 0.688·19-s + 1.96·21-s + 0.208·23-s − 1/5·25-s − 0.185·29-s + 1.43·31-s + 3.13·33-s − 1.52·35-s + 0.986·37-s − 2.40·39-s + 0.937·41-s − 0.304·43-s − 1.34·45-s + 0.437·47-s + 5/7·49-s + 0.840·51-s − 1.92·53-s − 2.42·55-s + 1.19·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.867001278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.867001278\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 7 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−13.8461199612, −13.1951938657, −12.5029481699, −12.1751926666, −11.7161913789, −11.6095709752, −11.1106389336, −10.4826222907, −9.73717745501, −9.42569735779, −9.17910842371, −8.56778807946, −8.02940186807, −7.83966715254, −7.65667253068, −6.94180128985, −6.43293561046, −5.62299186430, −4.80359686731, −4.43475021338, −3.89645626888, −3.37734431182, −2.77750842207, −2.07590271514, −1.09847792961,
1.09847792961, 2.07590271514, 2.77750842207, 3.37734431182, 3.89645626888, 4.43475021338, 4.80359686731, 5.62299186430, 6.43293561046, 6.94180128985, 7.65667253068, 7.83966715254, 8.02940186807, 8.56778807946, 9.17910842371, 9.42569735779, 9.73717745501, 10.4826222907, 11.1106389336, 11.6095709752, 11.7161913789, 12.1751926666, 12.5029481699, 13.1951938657, 13.8461199612
