| L(s) = 1 | − 2.34·2-s + 3.15·3-s + 3.50·4-s − 3.97·5-s − 7.39·6-s − 1.04·7-s − 3.52·8-s + 6.93·9-s + 9.31·10-s − 5.50·11-s + 11.0·12-s − 4.82·13-s + 2.44·14-s − 12.5·15-s + 1.25·16-s + 6.98·17-s − 16.2·18-s + 2.80·19-s − 13.9·20-s − 3.28·21-s + 12.9·22-s + 5.55·23-s − 11.1·24-s + 10.7·25-s + 11.3·26-s + 12.4·27-s − 3.65·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.81·3-s + 1.75·4-s − 1.77·5-s − 3.01·6-s − 0.394·7-s − 1.24·8-s + 2.31·9-s + 2.94·10-s − 1.65·11-s + 3.18·12-s − 1.33·13-s + 0.653·14-s − 3.23·15-s + 0.314·16-s + 1.69·17-s − 3.83·18-s + 0.643·19-s − 3.10·20-s − 0.717·21-s + 2.75·22-s + 1.15·23-s − 2.26·24-s + 2.15·25-s + 2.21·26-s + 2.38·27-s − 0.690·28-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 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 6.98T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 0.103T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 5.00T + 61T^{2} \) |
| 67 | \( 1 - 9.25T + 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 8.95T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63270629969870930001463316446, −7.47364861863579150767657628342, −6.95463349755450165500278498982, −5.25214106216988487292776095001, −4.47696203082000823776621625486, −3.23366599593409828378181635152, −3.13651731303346443697883474378, −2.29742360553416258419489106917, −1.06338283992664789354886648484, 0,
1.06338283992664789354886648484, 2.29742360553416258419489106917, 3.13651731303346443697883474378, 3.23366599593409828378181635152, 4.47696203082000823776621625486, 5.25214106216988487292776095001, 6.95463349755450165500278498982, 7.47364861863579150767657628342, 7.63270629969870930001463316446
