| L(s) = 1 | + 2-s + 0.581·3-s + 4-s + 1.63·5-s + 0.581·6-s + 2.77·7-s + 8-s − 2.66·9-s + 1.63·10-s + 5.30·11-s + 0.581·12-s − 0.644·13-s + 2.77·14-s + 0.949·15-s + 16-s + 5.86·17-s − 2.66·18-s + 19-s + 1.63·20-s + 1.61·21-s + 5.30·22-s − 1.45·23-s + 0.581·24-s − 2.33·25-s − 0.644·26-s − 3.29·27-s + 2.77·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.335·3-s + 0.5·4-s + 0.730·5-s + 0.237·6-s + 1.04·7-s + 0.353·8-s − 0.887·9-s + 0.516·10-s + 1.59·11-s + 0.167·12-s − 0.178·13-s + 0.741·14-s + 0.245·15-s + 0.250·16-s + 1.42·17-s − 0.627·18-s + 0.229·19-s + 0.365·20-s + 0.351·21-s + 1.13·22-s − 0.302·23-s + 0.118·24-s − 0.466·25-s − 0.126·26-s − 0.633·27-s + 0.524·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.486091668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.486091668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.581T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 + 0.644T + 13T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 + 6.21T + 31T^{2} \) |
| 37 | \( 1 - 0.975T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 - 5.95T + 59T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 + 3.10T + 67T^{2} \) |
| 71 | \( 1 - 0.985T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 7.50T + 79T^{2} \) |
| 83 | \( 1 - 4.43T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.60539751578531328171093265922, −7.31137205536425965582372228917, −6.08134685148368177378242116280, −5.75176441782354061736868066771, −5.22799358024325051880791164320, −4.06786141372589323727788933496, −3.73589078228531089704343296292, −2.63543232806421121935455499761, −1.90041510121531813032637386798, −1.12066002503445085802213356385,
1.12066002503445085802213356385, 1.90041510121531813032637386798, 2.63543232806421121935455499761, 3.73589078228531089704343296292, 4.06786141372589323727788933496, 5.22799358024325051880791164320, 5.75176441782354061736868066771, 6.08134685148368177378242116280, 7.31137205536425965582372228917, 7.60539751578531328171093265922
