| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s + 3·9-s + 2·11-s − 4·12-s − 4·16-s − 2·17-s − 6·18-s − 2·19-s − 4·22-s − 25-s − 4·27-s + 8·32-s − 4·33-s + 4·34-s + 6·36-s + 4·38-s − 2·43-s + 4·44-s + 8·48-s + 11·49-s + 2·50-s + 4·51-s + 8·54-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s + 9-s + 0.603·11-s − 1.15·12-s − 16-s − 0.485·17-s − 1.41·18-s − 0.458·19-s − 0.852·22-s − 1/5·25-s − 0.769·27-s + 1.41·32-s − 0.696·33-s + 0.685·34-s + 36-s + 0.648·38-s − 0.304·43-s + 0.603·44-s + 1.15·48-s + 11/7·49-s + 0.282·50-s + 0.560·51-s + 1.08·54-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.836427381447004052087018681826, −8.562174106312293436599005698498, −7.80605317251250760198226171715, −7.47848778675296589128891353641, −6.97556751348491276200701022616, −6.41173993136661837405396202601, −6.21053798963199366539375101013, −5.43607631650156514852377977246, −4.88619376922667033041566401202, −4.27866732192520382052996066610, −3.81591641511052084615077615666, −2.68139800535164168175122974103, −1.84871910742097553432780734662, −1.10084134564969783818306772778, 0,
1.10084134564969783818306772778, 1.84871910742097553432780734662, 2.68139800535164168175122974103, 3.81591641511052084615077615666, 4.27866732192520382052996066610, 4.88619376922667033041566401202, 5.43607631650156514852377977246, 6.21053798963199366539375101013, 6.41173993136661837405396202601, 6.97556751348491276200701022616, 7.47848778675296589128891353641, 7.80605317251250760198226171715, 8.562174106312293436599005698498, 8.836427381447004052087018681826
