| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 3·8-s + 3·9-s + 2·11-s + 2·12-s − 16-s − 4·17-s + 3·18-s + 2·22-s + 6·24-s − 6·25-s − 4·27-s + 5·32-s − 4·33-s − 4·34-s − 3·36-s − 4·41-s − 2·44-s + 2·48-s + 2·49-s − 6·50-s + 8·51-s − 4·54-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.06·8-s + 9-s + 0.603·11-s + 0.577·12-s − 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.426·22-s + 1.22·24-s − 6/5·25-s − 0.769·27-s + 0.883·32-s − 0.696·33-s − 0.685·34-s − 1/2·36-s − 0.624·41-s − 0.301·44-s + 0.288·48-s + 2/7·49-s − 0.848·50-s + 1.12·51-s − 0.544·54-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.707415080973846397836109340574, −8.975261543279013968212899152873, −8.875675328772527825961545316041, −7.965954059246732200439764911687, −7.46995816184461358823952634116, −6.70418908817771650365413799438, −6.33644802733938487268952260308, −5.84871034939782789631443011618, −5.31622482563829752699303340190, −4.70993425864659877425168996437, −4.20407903283329927801590362766, −3.73471833099679066824930451074, −2.75044853344397076993328420584, −1.54953448111991999222312546648, 0,
1.54953448111991999222312546648, 2.75044853344397076993328420584, 3.73471833099679066824930451074, 4.20407903283329927801590362766, 4.70993425864659877425168996437, 5.31622482563829752699303340190, 5.84871034939782789631443011618, 6.33644802733938487268952260308, 6.70418908817771650365413799438, 7.46995816184461358823952634116, 7.965954059246732200439764911687, 8.875675328772527825961545316041, 8.975261543279013968212899152873, 9.707415080973846397836109340574
