| L(s) = 1 | + 1.44·2-s + 3-s + 0.0986·4-s + 2.08·5-s + 1.44·6-s + 4.31·7-s − 2.75·8-s + 9-s + 3.02·10-s + 0.927·11-s + 0.0986·12-s + 0.332·13-s + 6.25·14-s + 2.08·15-s − 4.18·16-s + 6.67·17-s + 1.44·18-s − 4.65·19-s + 0.205·20-s + 4.31·21-s + 1.34·22-s − 23-s − 2.75·24-s − 0.641·25-s + 0.482·26-s + 27-s + 0.425·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.577·3-s + 0.0493·4-s + 0.933·5-s + 0.591·6-s + 1.63·7-s − 0.973·8-s + 0.333·9-s + 0.956·10-s + 0.279·11-s + 0.0284·12-s + 0.0922·13-s + 1.67·14-s + 0.539·15-s − 1.04·16-s + 1.61·17-s + 0.341·18-s − 1.06·19-s + 0.0460·20-s + 0.942·21-s + 0.286·22-s − 0.208·23-s − 0.562·24-s − 0.128·25-s + 0.0945·26-s + 0.192·27-s + 0.0804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.597766651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.597766651\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 11 | \( 1 - 0.927T + 11T^{2} \) |
| 13 | \( 1 - 0.332T + 13T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 + 1.09T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−9.081963468417426059704191155789, −8.355026508559586956117281469244, −7.73878306609414039106338683274, −6.55273220031425408327753147594, −5.69736102812519301775769919392, −5.09993709116232710406450707636, −4.31702650071014512838698115960, −3.45369022348525350489981561242, −2.30916634756999776355239623225, −1.44152321245688624182345666001,
1.44152321245688624182345666001, 2.30916634756999776355239623225, 3.45369022348525350489981561242, 4.31702650071014512838698115960, 5.09993709116232710406450707636, 5.69736102812519301775769919392, 6.55273220031425408327753147594, 7.73878306609414039106338683274, 8.355026508559586956117281469244, 9.081963468417426059704191155789
