| L(s) = 1 | + 2·7-s − 2·13-s + 2·19-s + 2·31-s − 2·43-s + 49-s + 2·61-s + 2·67-s − 4·91-s − 2·97-s + 4·103-s − 2·109-s + 127-s + 131-s + 4·133-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 | + 2·7-s − 2·13-s + 2·19-s + 2·31-s − 2·43-s + 49-s + 2·61-s + 2·67-s − 4·91-s − 2·97-s + 4·103-s − 2·109-s + 127-s + 131-s + 4·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.513027032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513027032\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.726849853760162435945790524760, −9.481643109819209187920328009155, −8.571117989757483871185316019410, −8.516328304255330979798076045165, −8.058916373388292310969097090446, −7.77888556726321503794629606575, −7.25221067118758668698812393630, −7.15616098528516266801275248004, −6.55212368187699617938367334150, −6.05150573135710332496011069713, −5.27200937995422452314448589512, −5.15345747603836662233310407075, −4.83784542716465231509154655959, −4.62705114421934455860040065344, −3.80244267575712987763138962422, −3.37350855651480880371050693834, −2.50054492117105069112874598586, −2.43891429590937091663326239392, −1.56505875568933193688038560825, −1.04320301703188116681147743593,
1.04320301703188116681147743593, 1.56505875568933193688038560825, 2.43891429590937091663326239392, 2.50054492117105069112874598586, 3.37350855651480880371050693834, 3.80244267575712987763138962422, 4.62705114421934455860040065344, 4.83784542716465231509154655959, 5.15345747603836662233310407075, 5.27200937995422452314448589512, 6.05150573135710332496011069713, 6.55212368187699617938367334150, 7.15616098528516266801275248004, 7.25221067118758668698812393630, 7.77888556726321503794629606575, 8.058916373388292310969097090446, 8.516328304255330979798076045165, 8.571117989757483871185316019410, 9.481643109819209187920328009155, 9.726849853760162435945790524760
