| L(s) = 1 | − 2.46·2-s + 2.55·3-s + 4.09·4-s + 0.103·5-s − 6.30·6-s + 0.172·7-s − 5.16·8-s + 3.53·9-s − 0.256·10-s + 0.565·11-s + 10.4·12-s + 2.15·13-s − 0.425·14-s + 0.265·15-s + 4.55·16-s − 1.75·17-s − 8.71·18-s − 0.748·19-s + 0.424·20-s + 0.440·21-s − 1.39·22-s + 8.39·23-s − 13.1·24-s − 4.98·25-s − 5.32·26-s + 1.35·27-s + 0.705·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 1.47·3-s + 2.04·4-s + 0.0464·5-s − 2.57·6-s + 0.0651·7-s − 1.82·8-s + 1.17·9-s − 0.0810·10-s + 0.170·11-s + 3.01·12-s + 0.598·13-s − 0.113·14-s + 0.0685·15-s + 1.13·16-s − 0.425·17-s − 2.05·18-s − 0.171·19-s + 0.0949·20-s + 0.0961·21-s − 0.297·22-s + 1.74·23-s − 2.69·24-s − 0.997·25-s − 1.04·26-s + 0.260·27-s + 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 2.55T + 3T^{2} \) |
| 5 | \( 1 - 0.103T + 5T^{2} \) |
| 7 | \( 1 - 0.172T + 7T^{2} \) |
| 11 | \( 1 - 0.565T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 + 1.75T + 17T^{2} \) |
| 19 | \( 1 + 0.748T + 19T^{2} \) |
| 23 | \( 1 - 8.39T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 - 4.50T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 5.33T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 0.701T + 71T^{2} \) |
| 73 | \( 1 + 5.45T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 1.56T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−7.87360780897902607488963185520, −7.16506757649282674813087514216, −6.57964389874238718869702660901, −5.65378704074210641916090013206, −4.44587019108519209808919053544, −3.46244577806651089535058408284, −2.89058238262482415565792820818, −1.88189057795960136658464537635, −1.48391538331764179749998811618, 0,
1.48391538331764179749998811618, 1.88189057795960136658464537635, 2.89058238262482415565792820818, 3.46244577806651089535058408284, 4.44587019108519209808919053544, 5.65378704074210641916090013206, 6.57964389874238718869702660901, 7.16506757649282674813087514216, 7.87360780897902607488963185520
