L(s) = 1 | − 2-s − 3-s + 4-s + 8·5-s + 6-s − 8-s − 2·9-s − 8·10-s − 12-s − 8·15-s + 16-s + 2·18-s − 2·19-s + 8·20-s + 2·23-s + 24-s + 38·25-s + 5·27-s + 10·29-s + 8·30-s − 32-s − 2·36-s + 2·38-s − 8·40-s + 8·43-s − 16·45-s − 2·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 3.57·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 2.52·10-s − 0.288·12-s − 2.06·15-s + 1/4·16-s + 0.471·18-s − 0.458·19-s + 1.78·20-s + 0.417·23-s + 0.204·24-s + 38/5·25-s + 0.962·27-s + 1.85·29-s + 1.46·30-s − 0.176·32-s − 1/3·36-s + 0.324·38-s − 1.26·40-s + 1.21·43-s − 2.38·45-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.266006194\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.266006194\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.900666531045489798996891080014, −8.336890921646432967363618675577, −7.892225159593170281658461026008, −6.73926084905849932341893089023, −6.67486851525401293685577568233, −6.41602681477419049736409749590, −5.70493789281341106083458944039, −5.70311644291868843897751049539, −4.89838836690724431377597203004, −4.80751556264464963514931464778, −3.35219753314436267860210658260, −2.55734550860524537659292391699, −2.45747388367035991238545338827, −1.62231881828892009952147969605, −1.03106032747078173831338248002,
1.03106032747078173831338248002, 1.62231881828892009952147969605, 2.45747388367035991238545338827, 2.55734550860524537659292391699, 3.35219753314436267860210658260, 4.80751556264464963514931464778, 4.89838836690724431377597203004, 5.70311644291868843897751049539, 5.70493789281341106083458944039, 6.41602681477419049736409749590, 6.67486851525401293685577568233, 6.73926084905849932341893089023, 7.892225159593170281658461026008, 8.336890921646432967363618675577, 8.900666531045489798996891080014