| L(s) = 1 | + 3.39·3-s − 3.99·5-s + 7-s + 8.51·9-s + 11-s − 13-s − 13.5·15-s − 0.471·17-s + 7.42·19-s + 3.39·21-s − 3.13·23-s + 10.9·25-s + 18.7·27-s + 0.836·29-s + 8.21·31-s + 3.39·33-s − 3.99·35-s − 10.4·37-s − 3.39·39-s − 4.29·41-s − 5.12·43-s − 33.9·45-s − 7.87·47-s + 49-s − 1.59·51-s + 10.7·53-s − 3.99·55-s + ⋯ |
| L(s) = 1 | + 1.95·3-s − 1.78·5-s + 0.377·7-s + 2.83·9-s + 0.301·11-s − 0.277·13-s − 3.49·15-s − 0.114·17-s + 1.70·19-s + 0.740·21-s − 0.652·23-s + 2.18·25-s + 3.60·27-s + 0.155·29-s + 1.47·31-s + 0.590·33-s − 0.674·35-s − 1.71·37-s − 0.543·39-s − 0.670·41-s − 0.781·43-s − 5.06·45-s − 1.14·47-s + 0.142·49-s − 0.223·51-s + 1.47·53-s − 0.538·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.325324764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.325324764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 3.99T + 5T^{2} \) |
| 17 | \( 1 + 0.471T + 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 - 0.836T + 29T^{2} \) |
| 31 | \( 1 - 8.21T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 0.786T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 - 4.21T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 - 0.624T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 - 5.92T + 89T^{2} \) |
| 97 | \( 1 + 2.95T + 97T^{2} \) |
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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
−8.423854818975783929026311017192, −7.896669950877711474183306632086, −7.23186310144707045355245550226, −6.82654401879253685460822923907, −5.03675450985733024956865359584, −4.40678189087951474416377247984, −3.50211570218513241435010858986, −3.31793113035944796016437313605, −2.17551736353612423281819182216, −1.00045760151462902103448476490,
1.00045760151462902103448476490, 2.17551736353612423281819182216, 3.31793113035944796016437313605, 3.50211570218513241435010858986, 4.40678189087951474416377247984, 5.03675450985733024956865359584, 6.82654401879253685460822923907, 7.23186310144707045355245550226, 7.896669950877711474183306632086, 8.423854818975783929026311017192
