| L(s) = 1 | − 1.50·2-s + 3-s + 0.262·4-s − 3.49·5-s − 1.50·6-s − 7-s + 2.61·8-s + 9-s + 5.25·10-s + 1.28·11-s + 0.262·12-s − 5.84·13-s + 1.50·14-s − 3.49·15-s − 4.45·16-s − 2.96·17-s − 1.50·18-s + 2.23·19-s − 0.917·20-s − 21-s − 1.93·22-s + 8.82·23-s + 2.61·24-s + 7.19·25-s + 8.78·26-s + 27-s − 0.262·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.577·3-s + 0.131·4-s − 1.56·5-s − 0.614·6-s − 0.377·7-s + 0.923·8-s + 0.333·9-s + 1.66·10-s + 0.387·11-s + 0.0758·12-s − 1.62·13-s + 0.402·14-s − 0.901·15-s − 1.11·16-s − 0.719·17-s − 0.354·18-s + 0.511·19-s − 0.205·20-s − 0.218·21-s − 0.411·22-s + 1.84·23-s + 0.533·24-s + 1.43·25-s + 1.72·26-s + 0.192·27-s − 0.0496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 + 5.84T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 0.553T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 6.46T + 43T^{2} \) |
| 47 | \( 1 + 0.680T + 47T^{2} \) |
| 53 | \( 1 + 4.80T + 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 + 8.82T + 61T^{2} \) |
| 67 | \( 1 - 7.42T + 67T^{2} \) |
| 71 | \( 1 + 5.90T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 6.66T + 89T^{2} \) |
| 97 | \( 1 - 0.433T + 97T^{2} \) |
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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
−7.57566293097283686250580909888, −7.20427643254017268013267623161, −6.61912626632691210112650104166, −5.01248727911842399667908432786, −4.68878869563866816445442176772, −3.78643045642101971950801487528, −3.11913988491013942369138211820, −2.14636606564365861152297299545, −0.894835547672909057171560622942, 0,
0.894835547672909057171560622942, 2.14636606564365861152297299545, 3.11913988491013942369138211820, 3.78643045642101971950801487528, 4.68878869563866816445442176772, 5.01248727911842399667908432786, 6.61912626632691210112650104166, 7.20427643254017268013267623161, 7.57566293097283686250580909888
