| L(s) = 1 | − 1.90·3-s − 3.63·5-s + 7-s + 0.619·9-s − 11-s + 13-s + 6.92·15-s − 3.76·17-s + 6.21·19-s − 1.90·21-s − 4.90·23-s + 8.24·25-s + 4.52·27-s − 9.49·29-s − 6.65·31-s + 1.90·33-s − 3.63·35-s − 9.21·37-s − 1.90·39-s − 0.741·41-s − 2.95·43-s − 2.25·45-s − 7.38·47-s + 49-s + 7.15·51-s + 1.74·53-s + 3.63·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s − 1.62·5-s + 0.377·7-s + 0.206·9-s − 0.301·11-s + 0.277·13-s + 1.78·15-s − 0.912·17-s + 1.42·19-s − 0.415·21-s − 1.02·23-s + 1.64·25-s + 0.871·27-s − 1.76·29-s − 1.19·31-s + 0.331·33-s − 0.615·35-s − 1.51·37-s − 0.304·39-s − 0.115·41-s − 0.450·43-s − 0.336·45-s − 1.07·47-s + 0.142·49-s + 1.00·51-s + 0.239·53-s + 0.490·55-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
| \(L(1)\) |
\(\approx\) |
\(0.3226198941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3226198941\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 - 6.21T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 + 9.49T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 0.741T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 + 7.38T + 47T^{2} \) |
| 53 | \( 1 - 1.74T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 + 9.53T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 5.31T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 3.99T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−8.389885638950493002789468547193, −7.48376906986090852796318820153, −7.22388531277944765529655117926, −6.16614147954629663826579587813, −5.37213441133760299574511278430, −4.79664900840494066179478189307, −3.88187064170773590163226346059, −3.26409674767775941754802196737, −1.75324870552607220471431575002, −0.33843276115774597643132156995,
0.33843276115774597643132156995, 1.75324870552607220471431575002, 3.26409674767775941754802196737, 3.88187064170773590163226346059, 4.79664900840494066179478189307, 5.37213441133760299574511278430, 6.16614147954629663826579587813, 7.22388531277944765529655117926, 7.48376906986090852796318820153, 8.389885638950493002789468547193
