| L(s) = 1 | + 2-s − 2.94·3-s + 4-s − 2.92·5-s − 2.94·6-s − 1.64·7-s + 8-s + 5.66·9-s − 2.92·10-s + 1.25·11-s − 2.94·12-s − 1.66·13-s − 1.64·14-s + 8.59·15-s + 16-s + 4.45·17-s + 5.66·18-s − 4.03·19-s − 2.92·20-s + 4.83·21-s + 1.25·22-s − 4.68·23-s − 2.94·24-s + 3.53·25-s − 1.66·26-s − 7.83·27-s − 1.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.69·3-s + 0.5·4-s − 1.30·5-s − 1.20·6-s − 0.620·7-s + 0.353·8-s + 1.88·9-s − 0.923·10-s + 0.379·11-s − 0.849·12-s − 0.461·13-s − 0.438·14-s + 2.21·15-s + 0.250·16-s + 1.08·17-s + 1.33·18-s − 0.924·19-s − 0.653·20-s + 1.05·21-s + 0.268·22-s − 0.977·23-s − 0.600·24-s + 0.706·25-s − 0.326·26-s − 1.50·27-s − 0.310·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.94T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 + 4.03T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 + 4.91T + 31T^{2} \) |
| 37 | \( 1 - 5.52T + 37T^{2} \) |
| 41 | \( 1 - 5.68T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 0.382T + 67T^{2} \) |
| 71 | \( 1 - 0.361T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 + 7.89T + 83T^{2} \) |
| 89 | \( 1 + 3.01T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.63245911733086185127194026910, −7.27195249455086692443056900580, −6.28297528185465108245583454989, −5.99601819134017085349847053201, −5.05441542526993445321420926362, −4.24477318744423620642299263583, −3.88733948076026008949129907380, −2.67366801108799123672317443183, −1.06311293982827942928534313600, 0,
1.06311293982827942928534313600, 2.67366801108799123672317443183, 3.88733948076026008949129907380, 4.24477318744423620642299263583, 5.05441542526993445321420926362, 5.99601819134017085349847053201, 6.28297528185465108245583454989, 7.27195249455086692443056900580, 7.63245911733086185127194026910
