L(s) = 1 | − 2.22·3-s − 0.143·5-s − 1.44·7-s + 1.92·9-s − 4.98·11-s + 13-s + 0.317·15-s − 2.87·17-s − 1.75·19-s + 3.20·21-s − 2.19·23-s − 4.97·25-s + 2.37·27-s − 29-s − 10.4·31-s + 11.0·33-s + 0.206·35-s − 3.76·37-s − 2.22·39-s + 9.68·41-s − 8.39·43-s − 0.276·45-s − 0.154·47-s − 4.91·49-s + 6.38·51-s + 1.92·53-s + 0.713·55-s + ⋯ |
L(s) = 1 | − 1.28·3-s − 0.0640·5-s − 0.545·7-s + 0.643·9-s − 1.50·11-s + 0.277·13-s + 0.0820·15-s − 0.697·17-s − 0.403·19-s + 0.699·21-s − 0.456·23-s − 0.995·25-s + 0.457·27-s − 0.185·29-s − 1.87·31-s + 1.92·33-s + 0.0349·35-s − 0.619·37-s − 0.355·39-s + 1.51·41-s − 1.28·43-s − 0.0411·45-s − 0.0225·47-s − 0.702·49-s + 0.894·51-s + 0.264·53-s + 0.0961·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1591545081\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1591545081\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 + 0.143T + 5T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 4.98T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 1.75T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 9.68T + 41T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 + 0.154T + 47T^{2} \) |
| 53 | \( 1 - 1.92T + 53T^{2} \) |
| 59 | \( 1 - 4.22T + 59T^{2} \) |
| 61 | \( 1 + 0.298T + 61T^{2} \) |
| 67 | \( 1 + 0.522T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 5.40T + 73T^{2} \) |
| 79 | \( 1 + 1.00T + 79T^{2} \) |
| 83 | \( 1 + 0.385T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.981584225173814579799831499228, −7.23226952044233179656225994783, −6.56533415432838785753815836900, −5.75687984211562368882494766216, −5.47417609368667665211483509427, −4.59272026029912408265017396170, −3.76875960619587348244139646366, −2.75017289902612941420918893970, −1.77431048359625991231096707122, −0.21276734828278822179572073195,
0.21276734828278822179572073195, 1.77431048359625991231096707122, 2.75017289902612941420918893970, 3.76875960619587348244139646366, 4.59272026029912408265017396170, 5.47417609368667665211483509427, 5.75687984211562368882494766216, 6.56533415432838785753815836900, 7.23226952044233179656225994783, 7.981584225173814579799831499228