| L(s) = 1 | − 13-s + 2·19-s + 25-s − 2·37-s + 3·43-s − 49-s + 61-s − 3·67-s − 73-s − 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 13-s + 2·19-s + 25-s − 2·37-s + 3·43-s − 49-s + 61-s − 3·67-s − 73-s − 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.296867386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296867386\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−8.952910772874176106280888074193, −8.943994934304339261617236592804, −8.655110398597480460679181888037, −7.87273262613030846417016401448, −7.63823020565407308784191310789, −7.34073364062009789488295651085, −6.96516226400799493069668125762, −6.77888406630811769064961417941, −5.83912396319886297117849941603, −5.79609977516594648501204724341, −5.46945916640562172902027589448, −4.70815618028868880075134397474, −4.66579629539841551547516813598, −4.17810278864933120319254758079, −3.31808992649296495843495233733, −3.18985694617474133857687924156, −2.76213216514709892984274457454, −2.05880563793082294983593070623, −1.50606362395437365325097187522, −0.76959534355878156740201573646,
0.76959534355878156740201573646, 1.50606362395437365325097187522, 2.05880563793082294983593070623, 2.76213216514709892984274457454, 3.18985694617474133857687924156, 3.31808992649296495843495233733, 4.17810278864933120319254758079, 4.66579629539841551547516813598, 4.70815618028868880075134397474, 5.46945916640562172902027589448, 5.79609977516594648501204724341, 5.83912396319886297117849941603, 6.77888406630811769064961417941, 6.96516226400799493069668125762, 7.34073364062009789488295651085, 7.63823020565407308784191310789, 7.87273262613030846417016401448, 8.655110398597480460679181888037, 8.943994934304339261617236592804, 8.952910772874176106280888074193
