L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 4·6-s + 3·9-s + 4·11-s + 4·12-s − 4·16-s − 4·17-s − 6·18-s − 10·19-s − 8·22-s + 4·27-s + 8·32-s + 8·33-s + 8·34-s + 6·36-s + 20·38-s − 16·41-s − 2·43-s + 8·44-s − 8·48-s − 5·49-s − 8·51-s − 8·54-s − 20·57-s − 20·59-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.20·11-s + 1.15·12-s − 16-s − 0.970·17-s − 1.41·18-s − 2.29·19-s − 1.70·22-s + 0.769·27-s + 1.41·32-s + 1.39·33-s + 1.37·34-s + 36-s + 3.24·38-s − 2.49·41-s − 0.304·43-s + 1.20·44-s − 1.15·48-s − 5/7·49-s − 1.12·51-s − 1.08·54-s − 2.64·57-s − 2.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−8.655074250265874594654667643843, −8.114757177440921419069207299350, −8.040086974208892692137412150811, −7.01259548848434300802989966251, −6.77287251257420576381216593643, −6.63552338173317415872093243656, −5.90404919386938898421911646081, −4.85692283621813202383375265351, −4.48018036005395779311676580836, −3.96102438209774458569462189574, −3.36957507180644347787302261531, −2.51201353173268436104607760745, −1.90319955654708104595670502735, −1.48067503408916751090348731726, 0,
1.48067503408916751090348731726, 1.90319955654708104595670502735, 2.51201353173268436104607760745, 3.36957507180644347787302261531, 3.96102438209774458569462189574, 4.48018036005395779311676580836, 4.85692283621813202383375265351, 5.90404919386938898421911646081, 6.63552338173317415872093243656, 6.77287251257420576381216593643, 7.01259548848434300802989966251, 8.040086974208892692137412150811, 8.114757177440921419069207299350, 8.655074250265874594654667643843