| L(s) = 1 | + 7-s − 2·9-s + 4·11-s − 12·23-s − 2·25-s + 2·29-s + 10·37-s + 4·43-s + 49-s + 10·53-s − 2·63-s − 4·67-s − 20·71-s + 4·77-s − 20·79-s − 5·81-s − 8·99-s + 4·107-s − 14·109-s + 4·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 2/3·9-s + 1.20·11-s − 2.50·23-s − 2/5·25-s + 0.371·29-s + 1.64·37-s + 0.609·43-s + 1/7·49-s + 1.37·53-s − 0.251·63-s − 0.488·67-s − 2.37·71-s + 0.455·77-s − 2.25·79-s − 5/9·81-s − 0.804·99-s + 0.386·107-s − 1.34·109-s + 0.376·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
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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
−7.65869261167450107823750782855, −7.45005912928402327913214471059, −6.85998409614637159802942519472, −6.21371174791133844116511861976, −6.00587253425096093343995602071, −5.76175796207076806488885877771, −5.08361012405941074736360249977, −4.39104688838354248863570972870, −4.07730036325899161420861564778, −3.82779271563122276431113592288, −2.90735366089804934483169027144, −2.51081508633609164424142946426, −1.78034092513983073922801136583, −1.16374911236534542788260164388, 0,
1.16374911236534542788260164388, 1.78034092513983073922801136583, 2.51081508633609164424142946426, 2.90735366089804934483169027144, 3.82779271563122276431113592288, 4.07730036325899161420861564778, 4.39104688838354248863570972870, 5.08361012405941074736360249977, 5.76175796207076806488885877771, 6.00587253425096093343995602071, 6.21371174791133844116511861976, 6.85998409614637159802942519472, 7.45005912928402327913214471059, 7.65869261167450107823750782855
