L(s) = 1 | − 2.71·2-s − 0.898·3-s + 5.39·4-s + 3.58·5-s + 2.44·6-s + 7-s − 9.21·8-s − 2.19·9-s − 9.75·10-s − 6.15·11-s − 4.84·12-s + 0.437·13-s − 2.71·14-s − 3.22·15-s + 14.2·16-s − 4.15·17-s + 5.96·18-s − 2.98·19-s + 19.3·20-s − 0.898·21-s + 16.7·22-s + 3.64·23-s + 8.28·24-s + 7.86·25-s − 1.18·26-s + 4.66·27-s + 5.39·28-s + ⋯ |
L(s) = 1 | − 1.92·2-s − 0.518·3-s + 2.69·4-s + 1.60·5-s + 0.997·6-s + 0.377·7-s − 3.25·8-s − 0.730·9-s − 3.08·10-s − 1.85·11-s − 1.39·12-s + 0.121·13-s − 0.726·14-s − 0.832·15-s + 3.57·16-s − 1.00·17-s + 1.40·18-s − 0.685·19-s + 4.32·20-s − 0.196·21-s + 3.56·22-s + 0.759·23-s + 1.69·24-s + 1.57·25-s − 0.233·26-s + 0.897·27-s + 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5365365666\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5365365666\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 3 | \( 1 + 0.898T + 3T^{2} \) |
| 5 | \( 1 - 3.58T + 5T^{2} \) |
| 11 | \( 1 + 6.15T + 11T^{2} \) |
| 13 | \( 1 - 0.437T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 + 2.98T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 0.0845T + 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 - 5.05T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 6.30T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 - 0.429T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 4.10T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 2.83T + 97T^{2} \) |
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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
−8.173244530233495172373269319969, −7.59031132853040117586472478217, −6.75422233915119218867267408474, −5.99800187417324469275117512294, −5.68735605032395513022170483499, −4.82565893693670964563875983204, −2.88401863853078972429438912118, −2.40017893929178812879971907137, −1.73611917311182210514439664467, −0.50468491968269553223459644335,
0.50468491968269553223459644335, 1.73611917311182210514439664467, 2.40017893929178812879971907137, 2.88401863853078972429438912118, 4.82565893693670964563875983204, 5.68735605032395513022170483499, 5.99800187417324469275117512294, 6.75422233915119218867267408474, 7.59031132853040117586472478217, 8.173244530233495172373269319969