| L(s) = 1 | − 2-s + 4-s − 8-s − 5·9-s − 8·13-s + 16-s − 6·17-s + 5·18-s + 8·26-s − 32-s + 6·34-s − 5·36-s + 4·37-s − 6·41-s − 10·49-s − 8·52-s + 12·53-s + 4·61-s + 64-s − 6·68-s + 5·72-s + 22·73-s − 4·74-s + 16·81-s + 6·82-s + 30·89-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 5/3·9-s − 2.21·13-s + 1/4·16-s − 1.45·17-s + 1.17·18-s + 1.56·26-s − 0.176·32-s + 1.02·34-s − 5/6·36-s + 0.657·37-s − 0.937·41-s − 1.42·49-s − 1.10·52-s + 1.64·53-s + 0.512·61-s + 1/8·64-s − 0.727·68-s + 0.589·72-s + 2.57·73-s − 0.464·74-s + 16/9·81-s + 0.662·82-s + 3.17·89-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 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
−10.81431448769284119152476376034, −9.919702445885858877180142258118, −9.447991916742081591691197868932, −9.107735829970857706936675027245, −8.316812877075225116243141130520, −8.082389583115066239879169681269, −7.29853134793764088505009214313, −6.71237324686107378754912317869, −6.21926731108516302667159437545, −5.20592148788576143134616479765, −4.98537291304699817306164733550, −3.80421461859967166401405341307, −2.65155040417235216240951299295, −2.32235507105178670692643054585, 0,
2.32235507105178670692643054585, 2.65155040417235216240951299295, 3.80421461859967166401405341307, 4.98537291304699817306164733550, 5.20592148788576143134616479765, 6.21926731108516302667159437545, 6.71237324686107378754912317869, 7.29853134793764088505009214313, 8.082389583115066239879169681269, 8.316812877075225116243141130520, 9.107735829970857706936675027245, 9.447991916742081591691197868932, 9.919702445885858877180142258118, 10.81431448769284119152476376034
