L(s) = 1 | + 0.296·2-s − 0.947·3-s − 1.91·4-s + 1.42·5-s − 0.280·6-s + 3.81·7-s − 1.15·8-s − 2.10·9-s + 0.420·10-s − 3.55·11-s + 1.81·12-s + 5.23·13-s + 1.13·14-s − 1.34·15-s + 3.48·16-s − 17-s − 0.622·18-s − 0.388·19-s − 2.71·20-s − 3.61·21-s − 1.05·22-s + 2.41·23-s + 1.09·24-s − 2.98·25-s + 1.55·26-s + 4.83·27-s − 7.30·28-s + ⋯ |
L(s) = 1 | + 0.209·2-s − 0.547·3-s − 0.956·4-s + 0.635·5-s − 0.114·6-s + 1.44·7-s − 0.409·8-s − 0.700·9-s + 0.133·10-s − 1.07·11-s + 0.523·12-s + 1.45·13-s + 0.302·14-s − 0.347·15-s + 0.870·16-s − 0.242·17-s − 0.146·18-s − 0.0891·19-s − 0.607·20-s − 0.789·21-s − 0.224·22-s + 0.502·23-s + 0.224·24-s − 0.596·25-s + 0.304·26-s + 0.930·27-s − 1.37·28-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
\(L(1)\) |
\(\approx\) |
\(1.407242953\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.407242953\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.296T + 2T^{2} \) |
| 3 | \( 1 + 0.947T + 3T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 19 | \( 1 + 0.388T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 + 0.981T + 67T^{2} \) |
| 71 | \( 1 - 6.59T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 + 1.55T + 89T^{2} \) |
| 97 | \( 1 - 7.52T + 97T^{2} \) |
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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
−10.16292432918992325819380874908, −8.836201179857904231100173806234, −8.544663240956798267150279805677, −7.64881439072417439786447660895, −6.17344084134335421959730822679, −5.50312241690851927825782262414, −4.96103451680392245664104627691, −3.93279585993839404049424500567, −2.48985900442327526078263325957, −0.968479810071920978024997125856,
0.968479810071920978024997125856, 2.48985900442327526078263325957, 3.93279585993839404049424500567, 4.96103451680392245664104627691, 5.50312241690851927825782262414, 6.17344084134335421959730822679, 7.64881439072417439786447660895, 8.544663240956798267150279805677, 8.836201179857904231100173806234, 10.16292432918992325819380874908