| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s − 3·9-s − 6·11-s + 2·12-s + 16-s − 6·17-s + 3·18-s + 10·19-s + 6·22-s − 2·24-s − 14·27-s − 32-s − 12·33-s + 6·34-s − 3·36-s − 10·38-s − 6·41-s − 8·43-s − 6·44-s + 2·48-s − 10·49-s − 12·51-s + 14·54-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 9-s − 1.80·11-s + 0.577·12-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 2.29·19-s + 1.27·22-s − 0.408·24-s − 2.69·27-s − 0.176·32-s − 2.08·33-s + 1.02·34-s − 1/2·36-s − 1.62·38-s − 0.937·41-s − 1.21·43-s − 0.904·44-s + 0.288·48-s − 1.42·49-s − 1.68·51-s + 1.90·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.447991916742081591691197868932, −8.972823812527392717881630254409, −8.316812877075225116243141130520, −8.242137217807967407949530158729, −7.62075816029599842618575360070, −7.29853134793764088505009214313, −6.48281153851554012248969831428, −5.85608845646892265745636218718, −5.20592148788576143134616479765, −4.87231762597053926583262812281, −3.55969126432596148811318362766, −3.06686786928646871283577398079, −2.65155040417235216240951299295, −1.87089602636661166790933941519, 0,
1.87089602636661166790933941519, 2.65155040417235216240951299295, 3.06686786928646871283577398079, 3.55969126432596148811318362766, 4.87231762597053926583262812281, 5.20592148788576143134616479765, 5.85608845646892265745636218718, 6.48281153851554012248969831428, 7.29853134793764088505009214313, 7.62075816029599842618575360070, 8.242137217807967407949530158729, 8.316812877075225116243141130520, 8.972823812527392717881630254409, 9.447991916742081591691197868932
