| L(s) = 1 | + 2.30·2-s + 0.298·3-s + 3.29·4-s − 3.47·5-s + 0.687·6-s − 4.18·7-s + 2.98·8-s − 2.91·9-s − 7.99·10-s + 4.06·11-s + 0.984·12-s − 5.70·13-s − 9.62·14-s − 1.03·15-s + 0.268·16-s − 17-s − 6.69·18-s − 3.10·19-s − 11.4·20-s − 1.24·21-s + 9.36·22-s + 8.75·23-s + 0.890·24-s + 7.05·25-s − 13.1·26-s − 1.76·27-s − 13.7·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.172·3-s + 1.64·4-s − 1.55·5-s + 0.280·6-s − 1.58·7-s + 1.05·8-s − 0.970·9-s − 2.52·10-s + 1.22·11-s + 0.284·12-s − 1.58·13-s − 2.57·14-s − 0.267·15-s + 0.0671·16-s − 0.242·17-s − 1.57·18-s − 0.711·19-s − 2.55·20-s − 0.272·21-s + 1.99·22-s + 1.82·23-s + 0.181·24-s + 1.41·25-s − 2.57·26-s − 0.339·27-s − 2.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 - 0.298T + 3T^{2} \) |
| 5 | \( 1 + 3.47T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 - 8.75T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 3.82T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.438399106731351393126731958695, −8.782803530085868062423759550980, −7.46783716682481555783963573366, −6.78824796687255351916201092112, −6.15799476389240550552691626158, −4.87659195788658754109102205016, −4.17533641394162965870248461665, −3.21850752286344195702396556098, −2.82156487144770476258545524931, 0,
2.82156487144770476258545524931, 3.21850752286344195702396556098, 4.17533641394162965870248461665, 4.87659195788658754109102205016, 6.15799476389240550552691626158, 6.78824796687255351916201092112, 7.46783716682481555783963573366, 8.782803530085868062423759550980, 9.438399106731351393126731958695
