| L(s) = 1 | + 3-s − 2·7-s − 2·21-s − 2·25-s − 27-s + 2·37-s + 49-s + 4·67-s + 2·73-s − 2·75-s − 81-s + 2·111-s − 121-s + 127-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·175-s + 179-s + ⋯ |
| L(s) = 1 | + 3-s − 2·7-s − 2·21-s − 2·25-s − 27-s + 2·37-s + 49-s + 4·67-s + 2·73-s − 2·75-s − 81-s + 2·111-s − 121-s + 127-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·175-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.033072113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033072113\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.921763969457085540197880810803, −9.219774604583693851293069149050, −9.191443691351216294935281831491, −8.317826138996527368174388897116, −8.224940826856757759559398109534, −7.78981566750885623079044498248, −7.44558115609293437294595820656, −6.71328791153383342956892919248, −6.61066182123351577581476996272, −6.18953032243392932579951806633, −5.61930233166320367103968150190, −5.46243394596558190526190514053, −4.62746884785173340421496838242, −3.98083067550842766143923055373, −3.80857314434344943208435299771, −3.24353399697834458977245882591, −2.97119765454596591758740963282, −2.24780809488665407879882799948, −2.03062225073722971842540154886, −0.69846380712620881234025110154,
0.69846380712620881234025110154, 2.03062225073722971842540154886, 2.24780809488665407879882799948, 2.97119765454596591758740963282, 3.24353399697834458977245882591, 3.80857314434344943208435299771, 3.98083067550842766143923055373, 4.62746884785173340421496838242, 5.46243394596558190526190514053, 5.61930233166320367103968150190, 6.18953032243392932579951806633, 6.61066182123351577581476996272, 6.71328791153383342956892919248, 7.44558115609293437294595820656, 7.78981566750885623079044498248, 8.224940826856757759559398109534, 8.317826138996527368174388897116, 9.191443691351216294935281831491, 9.219774604583693851293069149050, 9.921763969457085540197880810803
