| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·6-s − 9-s + 4·11-s − 2·12-s + 4·13-s + 16-s − 4·17-s + 2·18-s − 8·22-s + 2·23-s − 8·26-s + 6·27-s − 2·29-s − 12·31-s + 2·32-s − 8·33-s + 8·34-s − 36-s − 8·39-s − 10·41-s − 10·43-s + 4·44-s − 4·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s − 1/3·9-s + 1.20·11-s − 0.577·12-s + 1.10·13-s + 1/4·16-s − 0.970·17-s + 0.471·18-s − 1.70·22-s + 0.417·23-s − 1.56·26-s + 1.15·27-s − 0.371·29-s − 2.15·31-s + 0.353·32-s − 1.39·33-s + 1.37·34-s − 1/6·36-s − 1.28·39-s − 1.56·41-s − 1.52·43-s + 0.603·44-s − 0.589·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 22 T + 253 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.270139881745019125031883514938, −9.128559284586590646927824873308, −8.769028807961724135728149521020, −8.578577265116062520833555256577, −8.068675000134834534207522684220, −7.49688846678257455948203647142, −7.07032931249808046852500496355, −6.48710691991034864115711384304, −6.32239009407352489369701850806, −5.92735749950727992255952659781, −5.32373646266677854598255942558, −5.01309159538756827941617182077, −4.35728407186587551123911184795, −3.73311687125108738249894829233, −3.38296902378199131051438889362, −2.61431899139832358370219492633, −1.51532476099228957169806701976, −1.38564124011964026382221160509, 0, 0,
1.38564124011964026382221160509, 1.51532476099228957169806701976, 2.61431899139832358370219492633, 3.38296902378199131051438889362, 3.73311687125108738249894829233, 4.35728407186587551123911184795, 5.01309159538756827941617182077, 5.32373646266677854598255942558, 5.92735749950727992255952659781, 6.32239009407352489369701850806, 6.48710691991034864115711384304, 7.07032931249808046852500496355, 7.49688846678257455948203647142, 8.068675000134834534207522684220, 8.578577265116062520833555256577, 8.769028807961724135728149521020, 9.128559284586590646927824873308, 9.270139881745019125031883514938
