| L(s) = 1 | + 2-s − 1.75·3-s + 4-s + 1.48·5-s − 1.75·6-s + 4.69·7-s + 8-s + 0.0778·9-s + 1.48·10-s − 3.31·11-s − 1.75·12-s + 5.09·13-s + 4.69·14-s − 2.61·15-s + 16-s − 2.48·17-s + 0.0778·18-s − 0.550·19-s + 1.48·20-s − 8.22·21-s − 3.31·22-s + 5.58·23-s − 1.75·24-s − 2.78·25-s + 5.09·26-s + 5.12·27-s + 4.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.01·3-s + 0.5·4-s + 0.665·5-s − 0.716·6-s + 1.77·7-s + 0.353·8-s + 0.0259·9-s + 0.470·10-s − 0.999·11-s − 0.506·12-s + 1.41·13-s + 1.25·14-s − 0.673·15-s + 0.250·16-s − 0.602·17-s + 0.0183·18-s − 0.126·19-s + 0.332·20-s − 1.79·21-s − 0.706·22-s + 1.16·23-s − 0.358·24-s − 0.557·25-s + 1.00·26-s + 0.986·27-s + 0.886·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.029379915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.029379915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 + 0.550T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 8.67T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 - 5.15T + 41T^{2} \) |
| 43 | \( 1 - 5.66T + 43T^{2} \) |
| 47 | \( 1 + 0.641T + 47T^{2} \) |
| 53 | \( 1 + 2.92T + 53T^{2} \) |
| 59 | \( 1 + 2.52T + 59T^{2} \) |
| 61 | \( 1 + 0.801T + 61T^{2} \) |
| 67 | \( 1 - 0.653T + 67T^{2} \) |
| 71 | \( 1 - 3.08T + 71T^{2} \) |
| 73 | \( 1 - 1.58T + 73T^{2} \) |
| 79 | \( 1 + 5.97T + 79T^{2} \) |
| 83 | \( 1 - 5.53T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−8.396595540611856060768120098587, −7.65664507559597233588503380243, −6.69558403913959035874301116963, −6.00202394654698648653110368816, −5.37372970419180840667158072417, −4.93093621798640657389826714826, −4.19715167017643150165017649472, −2.89989546653284747630819658272, −1.93669944003592167386096469711, −1.00737709021645256272658501606,
1.00737709021645256272658501606, 1.93669944003592167386096469711, 2.89989546653284747630819658272, 4.19715167017643150165017649472, 4.93093621798640657389826714826, 5.37372970419180840667158072417, 6.00202394654698648653110368816, 6.69558403913959035874301116963, 7.65664507559597233588503380243, 8.396595540611856060768120098587
