| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 4·6-s + 3·9-s + 4·12-s − 2·13-s − 4·16-s − 6·18-s + 4·26-s + 4·27-s − 6·31-s + 8·32-s + 6·36-s − 4·37-s − 4·39-s − 16·41-s − 2·43-s − 8·48-s − 5·49-s − 4·52-s + 8·53-s − 8·54-s + 12·62-s − 8·64-s + 6·67-s − 16·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.15·12-s − 0.554·13-s − 16-s − 1.41·18-s + 0.784·26-s + 0.769·27-s − 1.07·31-s + 1.41·32-s + 36-s − 0.657·37-s − 0.640·39-s − 2.49·41-s − 0.304·43-s − 1.15·48-s − 5/7·49-s − 0.554·52-s + 1.09·53-s − 1.08·54-s + 1.52·62-s − 64-s + 0.733·67-s − 1.89·71-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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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, −7.74916178315119539489350856070, −7.30514032256911612413843551960, −6.77287251257420576381216593643, −6.65396759238287778208236419506, −5.58318025222206456893422400511, −5.14493093108369759562477643348, −4.48018036005395779311676580836, −3.89570536316911001261511296620, −3.25692776068793976659434863504, −2.64453939621087097050043751453, −1.90319955654708104595670502735, −1.43286730840512162749271730076, 0,
1.43286730840512162749271730076, 1.90319955654708104595670502735, 2.64453939621087097050043751453, 3.25692776068793976659434863504, 3.89570536316911001261511296620, 4.48018036005395779311676580836, 5.14493093108369759562477643348, 5.58318025222206456893422400511, 6.65396759238287778208236419506, 6.77287251257420576381216593643, 7.30514032256911612413843551960, 7.74916178315119539489350856070, 8.114757177440921419069207299350, 8.655074250265874594654667643843
