| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 5-s + 9·6-s − 3·8-s + 6·9-s + 3·10-s − 6·11-s − 12·12-s + 3·15-s + 3·16-s + 6·17-s − 18·18-s + 6·19-s − 4·20-s + 18·22-s − 6·23-s + 9·24-s − 9·27-s − 9·30-s + 6·31-s − 6·32-s + 18·33-s − 18·34-s + 24·36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 0.447·5-s + 3.67·6-s − 1.06·8-s + 2·9-s + 0.948·10-s − 1.80·11-s − 3.46·12-s + 0.774·15-s + 3/4·16-s + 1.45·17-s − 4.24·18-s + 1.37·19-s − 0.894·20-s + 3.83·22-s − 1.25·23-s + 1.83·24-s − 1.73·27-s − 1.64·30-s + 1.07·31-s − 1.06·32-s + 3.13·33-s − 3.08·34-s + 4·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−10.02602025289590951429925809664, −9.954434108767726224492867325613, −9.605916060702187968440168280532, −8.925909160852142324607829471465, −8.229660901280353154290682578055, −8.067588576418741997280668476340, −7.72174381128077248587751131451, −7.43422218943838183854342870707, −6.72577597949370955525416273532, −6.36079365394078231659565467589, −5.57889991250662489813909274002, −5.50162065303228152018525141007, −4.86248568477507620396039223219, −4.44823698710539668065695059809, −3.19925851719562174403893913931, −3.14761293328571175215733512142, −1.66028423997320428946064271532, −1.24176101119156301268555931664, 0, 0,
1.24176101119156301268555931664, 1.66028423997320428946064271532, 3.14761293328571175215733512142, 3.19925851719562174403893913931, 4.44823698710539668065695059809, 4.86248568477507620396039223219, 5.50162065303228152018525141007, 5.57889991250662489813909274002, 6.36079365394078231659565467589, 6.72577597949370955525416273532, 7.43422218943838183854342870707, 7.72174381128077248587751131451, 8.067588576418741997280668476340, 8.229660901280353154290682578055, 8.925909160852142324607829471465, 9.605916060702187968440168280532, 9.954434108767726224492867325613, 10.02602025289590951429925809664
