| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s + 6·11-s − 6·12-s + 2·13-s − 4·14-s + 5·16-s + 3·17-s − 6·18-s − 4·21-s − 12·22-s − 5·23-s + 8·24-s − 10·25-s − 4·26-s − 4·27-s + 6·28-s + 8·31-s − 6·32-s − 12·33-s − 6·34-s + 9·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 1.80·11-s − 1.73·12-s + 0.554·13-s − 1.06·14-s + 5/4·16-s + 0.727·17-s − 1.41·18-s − 0.872·21-s − 2.55·22-s − 1.04·23-s + 1.63·24-s − 2·25-s − 0.784·26-s − 0.769·27-s + 1.13·28-s + 1.43·31-s − 1.06·32-s − 2.08·33-s − 1.02·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64545156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64545156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 103 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 148 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 17 T + 190 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.62631832822417467090955440465, −7.51904360147539411559254789457, −6.80134271561677692582631559620, −6.76436811596664401302268813127, −6.26437069808762751728525774063, −6.11009550649154300352822527504, −5.66203240795335617103849990702, −5.56340933589476767118856649691, −4.68312167242100935058170540725, −4.64714249185143162519713610086, −3.99088063152956238308411010891, −3.89154438857879761822429097782, −3.10076682835162918317876340505, −3.01128590005100098459338882481, −1.89061483785317462751937237823, −1.80650834061989639068513376324, −1.27238555324740662187355618836, −1.21640907719390999594714792637, 0, 0,
1.21640907719390999594714792637, 1.27238555324740662187355618836, 1.80650834061989639068513376324, 1.89061483785317462751937237823, 3.01128590005100098459338882481, 3.10076682835162918317876340505, 3.89154438857879761822429097782, 3.99088063152956238308411010891, 4.64714249185143162519713610086, 4.68312167242100935058170540725, 5.56340933589476767118856649691, 5.66203240795335617103849990702, 6.11009550649154300352822527504, 6.26437069808762751728525774063, 6.76436811596664401302268813127, 6.80134271561677692582631559620, 7.51904360147539411559254789457, 7.62631832822417467090955440465
