| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 3·9-s + 4·12-s − 4·16-s + 6·18-s + 4·27-s + 20·31-s − 8·32-s + 6·36-s − 8·37-s + 20·41-s − 8·43-s − 8·48-s + 10·49-s − 20·53-s + 8·54-s + 40·62-s − 8·64-s + 24·67-s − 8·71-s − 16·74-s + 28·79-s + 5·81-s + 40·82-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 9-s + 1.15·12-s − 16-s + 1.41·18-s + 0.769·27-s + 3.59·31-s − 1.41·32-s + 36-s − 1.31·37-s + 3.12·41-s − 1.21·43-s − 1.15·48-s + 10/7·49-s − 2.74·53-s + 1.08·54-s + 5.08·62-s − 64-s + 2.93·67-s − 0.949·71-s − 1.85·74-s + 3.15·79-s + 5/9·81-s + 4.41·82-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
| \(L(1)\) |
\(\approx\) |
\(6.023851464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.023851464\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.17280213434441775729354164527, −10.46558207188909209278887373858, −9.862013650422922002899832201568, −9.662488699462513012352011175620, −9.138990688243248336757955051629, −8.640995296333488094638450261966, −8.129832662337429748606819928419, −7.937331385299339575120187278410, −7.25484778424901926075178841069, −6.58055249715388620260134079843, −6.50607490609173362501205439226, −5.83440828607214858299831450367, −5.20591690215189884082556029488, −4.63689501371715664178488111216, −4.36377315028797765063199726698, −3.72465792120125972765801956572, −3.22959532377526981731033881577, −2.60573082288179315085877908054, −2.30106681279212984081645907654, −1.12033754783572824717961723555,
1.12033754783572824717961723555, 2.30106681279212984081645907654, 2.60573082288179315085877908054, 3.22959532377526981731033881577, 3.72465792120125972765801956572, 4.36377315028797765063199726698, 4.63689501371715664178488111216, 5.20591690215189884082556029488, 5.83440828607214858299831450367, 6.50607490609173362501205439226, 6.58055249715388620260134079843, 7.25484778424901926075178841069, 7.937331385299339575120187278410, 8.129832662337429748606819928419, 8.640995296333488094638450261966, 9.138990688243248336757955051629, 9.662488699462513012352011175620, 9.862013650422922002899832201568, 10.46558207188909209278887373858, 11.17280213434441775729354164527
