| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·15-s + 12·17-s − 8·19-s + 3·25-s + 4·27-s − 16·31-s − 6·45-s + 2·49-s + 24·51-s − 16·57-s − 24·59-s − 4·61-s + 8·67-s + 20·73-s + 6·75-s + 16·79-s + 5·81-s − 24·85-s − 32·93-s + 16·95-s − 12·101-s − 32·103-s − 24·107-s + 10·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s − 1.03·15-s + 2.91·17-s − 1.83·19-s + 3/5·25-s + 0.769·27-s − 2.87·31-s − 0.894·45-s + 2/7·49-s + 3.36·51-s − 2.11·57-s − 3.12·59-s − 0.512·61-s + 0.977·67-s + 2.34·73-s + 0.692·75-s + 1.80·79-s + 5/9·81-s − 2.60·85-s − 3.31·93-s + 1.64·95-s − 1.19·101-s − 3.15·103-s − 2.32·107-s + 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.267829368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.267829368\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.460152976981706593284473992816, −8.018598223541456384868013921902, −7.85094560334792266325663344704, −7.56647856111963016942641309268, −7.32103868569831823304225124888, −6.72602237547330614174529718153, −6.49953479370163999640792460265, −5.92319434153301586199589305131, −5.57127623534603184219590009146, −5.00174978450755419537210465904, −4.93889407390658293860696851379, −4.01008805816627195268461206719, −3.95722762751561373908472329755, −3.58178377958223990297484956940, −3.26603766495715402476871054369, −2.71150527587835831433144110884, −2.30682750869853301897391270928, −1.55613390707426015664982458515, −1.37372377307192680778531789547, −0.38293732144677557968102448870,
0.38293732144677557968102448870, 1.37372377307192680778531789547, 1.55613390707426015664982458515, 2.30682750869853301897391270928, 2.71150527587835831433144110884, 3.26603766495715402476871054369, 3.58178377958223990297484956940, 3.95722762751561373908472329755, 4.01008805816627195268461206719, 4.93889407390658293860696851379, 5.00174978450755419537210465904, 5.57127623534603184219590009146, 5.92319434153301586199589305131, 6.49953479370163999640792460265, 6.72602237547330614174529718153, 7.32103868569831823304225124888, 7.56647856111963016942641309268, 7.85094560334792266325663344704, 8.018598223541456384868013921902, 8.460152976981706593284473992816
