| L(s) = 1 | − 3-s − 4·4-s + 8·7-s − 2·9-s + 4·12-s − 2·13-s + 12·16-s − 8·19-s − 8·21-s + 5·27-s − 32·28-s − 8·31-s + 8·36-s − 4·37-s + 2·39-s + 14·43-s − 12·48-s + 34·49-s + 8·52-s + 8·57-s − 2·61-s − 16·63-s − 32·64-s − 28·67-s + 8·73-s + 32·76-s + 22·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s + 3.02·7-s − 2/3·9-s + 1.15·12-s − 0.554·13-s + 3·16-s − 1.83·19-s − 1.74·21-s + 0.962·27-s − 6.04·28-s − 1.43·31-s + 4/3·36-s − 0.657·37-s + 0.320·39-s + 2.13·43-s − 1.73·48-s + 34/7·49-s + 1.10·52-s + 1.05·57-s − 0.256·61-s − 2.01·63-s − 4·64-s − 3.42·67-s + 0.936·73-s + 3.67·76-s + 2.47·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.109622323430795384540902637182, −7.58470653085664427086815596244, −7.44313533967700149106082663018, −6.43995160346575362242193439929, −5.84441229372916213818129575211, −5.50324850311908281715674980863, −5.07836611320697081751291448911, −4.79815352058635420780275090547, −4.33348131529579942321198409567, −4.15253391774556813403352263543, −3.35745100328651024034839394504, −2.32784842593438550687981568602, −1.76397611793551290887183908518, −0.989438760515621806827446532171, 0,
0.989438760515621806827446532171, 1.76397611793551290887183908518, 2.32784842593438550687981568602, 3.35745100328651024034839394504, 4.15253391774556813403352263543, 4.33348131529579942321198409567, 4.79815352058635420780275090547, 5.07836611320697081751291448911, 5.50324850311908281715674980863, 5.84441229372916213818129575211, 6.43995160346575362242193439929, 7.44313533967700149106082663018, 7.58470653085664427086815596244, 8.109622323430795384540902637182
