| L(s) = 1 | + 1.35·2-s − 1.79·3-s − 0.166·4-s + 1.77·5-s − 2.43·6-s + 2.41·7-s − 2.93·8-s + 0.235·9-s + 2.40·10-s − 4.25·11-s + 0.300·12-s − 4.33·13-s + 3.26·14-s − 3.19·15-s − 3.63·16-s + 17-s + 0.318·18-s − 0.153·19-s − 0.296·20-s − 4.34·21-s − 5.75·22-s + 4.64·23-s + 5.27·24-s − 1.84·25-s − 5.87·26-s + 4.97·27-s − 0.403·28-s + ⋯ |
| L(s) = 1 | + 0.957·2-s − 1.03·3-s − 0.0834·4-s + 0.793·5-s − 0.994·6-s + 0.912·7-s − 1.03·8-s + 0.0783·9-s + 0.760·10-s − 1.28·11-s + 0.0866·12-s − 1.20·13-s + 0.873·14-s − 0.824·15-s − 0.909·16-s + 0.242·17-s + 0.0750·18-s − 0.0352·19-s − 0.0662·20-s − 0.947·21-s − 1.22·22-s + 0.968·23-s + 1.07·24-s − 0.369·25-s − 1.15·26-s + 0.957·27-s − 0.0761·28-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 | \( 1 - 1.35T + 2T^{2} \) |
| 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 4.25T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 19 | \( 1 + 0.153T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 + 2.97T + 29T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.456T + 47T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 0.309T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + 3.81T + 73T^{2} \) |
| 79 | \( 1 + 4.58T + 79T^{2} \) |
| 83 | \( 1 - 4.13T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−9.781774512268193474251741761715, −8.760656248070404481974454631239, −7.74455930238718215076506888022, −6.69078339318917569036721918753, −5.63958280666452974072637957929, −5.15659529714884165733686940878, −4.81357956400948216824582865592, −3.23626276081557395490977497786, −2.04327356620007429191694169137, 0,
2.04327356620007429191694169137, 3.23626276081557395490977497786, 4.81357956400948216824582865592, 5.15659529714884165733686940878, 5.63958280666452974072637957929, 6.69078339318917569036721918753, 7.74455930238718215076506888022, 8.760656248070404481974454631239, 9.781774512268193474251741761715
