| L(s) = 1 | − 2·2-s + 2·4-s + 9-s − 2·13-s − 4·16-s − 4·17-s − 2·18-s + 4·26-s + 20·29-s + 8·32-s + 8·34-s + 2·36-s − 4·37-s − 16·41-s − 5·49-s − 4·52-s + 8·53-s − 40·58-s + 14·61-s − 8·64-s − 8·68-s + 28·73-s + 8·74-s + 81-s + 32·82-s − 34·97-s + 10·98-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1/3·9-s − 0.554·13-s − 16-s − 0.970·17-s − 0.471·18-s + 0.784·26-s + 3.71·29-s + 1.41·32-s + 1.37·34-s + 1/3·36-s − 0.657·37-s − 2.49·41-s − 5/7·49-s − 0.554·52-s + 1.09·53-s − 5.25·58-s + 1.79·61-s − 64-s − 0.970·68-s + 3.27·73-s + 0.929·74-s + 1/9·81-s + 3.53·82-s − 3.45·97-s + 1.01·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6549206182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6549206182\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.903619965777680376182050580499, −9.052471577485418050377330134019, −8.655074250265874594654667643843, −8.114757177440921419069207299350, −8.105315307318281005924759006469, −6.96660723932071234204269144809, −6.77287251257420576381216593643, −6.59754059703926529506773066348, −5.45352832460903594958062456545, −4.78985931636144963960080177818, −4.48018036005395779311676580836, −3.48045838663060024632885892599, −2.58914107657822994764341775063, −1.90319955654708104595670502735, −0.77482341387654784778243330666,
0.77482341387654784778243330666, 1.90319955654708104595670502735, 2.58914107657822994764341775063, 3.48045838663060024632885892599, 4.48018036005395779311676580836, 4.78985931636144963960080177818, 5.45352832460903594958062456545, 6.59754059703926529506773066348, 6.77287251257420576381216593643, 6.96660723932071234204269144809, 8.105315307318281005924759006469, 8.114757177440921419069207299350, 8.655074250265874594654667643843, 9.052471577485418050377330134019, 9.903619965777680376182050580499
