| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 3·9-s + 4·11-s − 4·12-s − 4·16-s + 4·17-s + 6·18-s − 10·19-s + 8·22-s − 4·27-s − 8·32-s − 8·33-s + 8·34-s + 6·36-s − 20·38-s − 16·41-s + 2·43-s + 8·44-s + 8·48-s − 5·49-s − 8·51-s − 8·54-s + 20·57-s − 20·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.20·11-s − 1.15·12-s − 16-s + 0.970·17-s + 1.41·18-s − 2.29·19-s + 1.70·22-s − 0.769·27-s − 1.41·32-s − 1.39·33-s + 1.37·34-s + 36-s − 3.24·38-s − 2.49·41-s + 0.304·43-s + 1.20·44-s + 1.15·48-s − 5/7·49-s − 1.12·51-s − 1.08·54-s + 2.64·57-s − 2.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−8.546309213405963304177966155831, −7.87914659835863207294698651709, −7.26683856545720539350458472989, −6.79375443883521821659151920822, −6.35078788674859422957686456838, −6.05569215012514529764032801313, −5.76958242735491544916508279271, −4.92993401446228872799172657878, −4.62309319523854591570832526017, −4.30861693097863502278050736122, −3.51380796888682305453962490670, −3.24466149791278329969031792942, −2.09699607926453765135786428693, −1.48244217054512001330990913109, 0,
1.48244217054512001330990913109, 2.09699607926453765135786428693, 3.24466149791278329969031792942, 3.51380796888682305453962490670, 4.30861693097863502278050736122, 4.62309319523854591570832526017, 4.92993401446228872799172657878, 5.76958242735491544916508279271, 6.05569215012514529764032801313, 6.35078788674859422957686456838, 6.79375443883521821659151920822, 7.26683856545720539350458472989, 7.87914659835863207294698651709, 8.546309213405963304177966155831
