| L(s) = 1 | + 2-s + 4-s + 8-s − 5·9-s + 2·13-s + 16-s − 5·18-s − 10·25-s + 2·26-s + 32-s − 5·36-s − 14·37-s + 6·41-s + 49-s − 10·50-s + 2·52-s − 24·53-s − 2·61-s + 64-s − 5·72-s + 22·73-s − 14·74-s + 16·81-s + 6·82-s − 36·89-s + 34·97-s + 98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 5/3·9-s + 0.554·13-s + 1/4·16-s − 1.17·18-s − 2·25-s + 0.392·26-s + 0.176·32-s − 5/6·36-s − 2.30·37-s + 0.937·41-s + 1/7·49-s − 1.41·50-s + 0.277·52-s − 3.29·53-s − 0.256·61-s + 1/8·64-s − 0.589·72-s + 2.57·73-s − 1.62·74-s + 16/9·81-s + 0.662·82-s − 3.81·89-s + 3.45·97-s + 0.101·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−8.474366624001991302416802379216, −8.357542398475312637001956914796, −7.63380170955425435565378568582, −7.37908920927623601208890863309, −6.49972537294879181773432831310, −6.10743185313712551583625613621, −5.92908447901902110795424240562, −5.06509234546123458101387314677, −5.02890614745559905024763859198, −3.94823754743172236057368898190, −3.60858467418748472558474969025, −3.03438118997979637652891820124, −2.33000973337707432224237511883, −1.60256072097181437267558889267, 0,
1.60256072097181437267558889267, 2.33000973337707432224237511883, 3.03438118997979637652891820124, 3.60858467418748472558474969025, 3.94823754743172236057368898190, 5.02890614745559905024763859198, 5.06509234546123458101387314677, 5.92908447901902110795424240562, 6.10743185313712551583625613621, 6.49972537294879181773432831310, 7.37908920927623601208890863309, 7.63380170955425435565378568582, 8.357542398475312637001956914796, 8.474366624001991302416802379216
