| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·7-s − 3·8-s − 3·9-s − 2·10-s + 11-s − 3·13-s − 3·14-s − 16-s − 3·18-s + 2·20-s + 22-s + 2·25-s − 3·26-s + 3·28-s − 3·31-s + 5·32-s + 6·35-s + 3·36-s + 6·40-s − 12·43-s − 44-s + 6·45-s + 12·47-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.13·7-s − 1.06·8-s − 9-s − 0.632·10-s + 0.301·11-s − 0.832·13-s − 0.801·14-s − 1/4·16-s − 0.707·18-s + 0.447·20-s + 0.213·22-s + 2/5·25-s − 0.588·26-s + 0.566·28-s − 0.538·31-s + 0.883·32-s + 1.01·35-s + 1/2·36-s + 0.948·40-s − 1.82·43-s − 0.150·44-s + 0.894·45-s + 1.75·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
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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
−10.96239475977261265437401831560, −10.21906533561043611547522146307, −9.701522321659404141328852604713, −9.113276727770291027943770721679, −8.701422134977613715299200488865, −8.084701980984663328503526362712, −7.34764711822666863228185593616, −6.72738373618797346492226426834, −6.04918393150238494247516438834, −5.45896305487914995573018765751, −4.74626267474400246663753655462, −3.96539536481289814528370580310, −3.35221128394077244011021395545, −2.69975189927794208387567623356, 0,
2.69975189927794208387567623356, 3.35221128394077244011021395545, 3.96539536481289814528370580310, 4.74626267474400246663753655462, 5.45896305487914995573018765751, 6.04918393150238494247516438834, 6.72738373618797346492226426834, 7.34764711822666863228185593616, 8.084701980984663328503526362712, 8.701422134977613715299200488865, 9.113276727770291027943770721679, 9.701522321659404141328852604713, 10.21906533561043611547522146307, 10.96239475977261265437401831560
