| L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 4·8-s + 9-s − 2·11-s + 2·14-s + 5·16-s + 2·18-s − 4·22-s − 16·23-s + 6·25-s + 3·28-s − 12·29-s + 6·32-s + 3·36-s − 12·37-s + 8·43-s − 6·44-s − 32·46-s + 49-s + 12·50-s + 4·53-s + 4·56-s − 24·58-s + 63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s + 1/3·9-s − 0.603·11-s + 0.534·14-s + 5/4·16-s + 0.471·18-s − 0.852·22-s − 3.33·23-s + 6/5·25-s + 0.566·28-s − 2.22·29-s + 1.06·32-s + 1/2·36-s − 1.97·37-s + 1.21·43-s − 0.904·44-s − 4.71·46-s + 1/7·49-s + 1.69·50-s + 0.549·53-s + 0.534·56-s − 3.15·58-s + 0.125·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.56853928475057757956492711754, −7.06196128452260537589494409619, −7.01388368279354966786295768024, −6.04350352134711634862151937108, −5.78641126273592614924413513429, −5.68884256231510340821014445655, −4.99001449691342549407846970736, −4.50182617970418156245344086250, −4.10732223636165202223499092609, −3.72515787941803042519247102042, −3.18439835936289363592396358072, −2.48203239562124190952326585155, −1.95397093408402137319315879397, −1.53287618374872470103273275153, 0,
1.53287618374872470103273275153, 1.95397093408402137319315879397, 2.48203239562124190952326585155, 3.18439835936289363592396358072, 3.72515787941803042519247102042, 4.10732223636165202223499092609, 4.50182617970418156245344086250, 4.99001449691342549407846970736, 5.68884256231510340821014445655, 5.78641126273592614924413513429, 6.04350352134711634862151937108, 7.01388368279354966786295768024, 7.06196128452260537589494409619, 7.56853928475057757956492711754
