| L(s) = 1 | − 2-s − 2.56·3-s + 4-s + 3.62·5-s + 2.56·6-s + 4.81·7-s − 8-s + 3.58·9-s − 3.62·10-s + 0.483·11-s − 2.56·12-s − 5.40·13-s − 4.81·14-s − 9.29·15-s + 16-s − 5.64·17-s − 3.58·18-s − 19-s + 3.62·20-s − 12.3·21-s − 0.483·22-s − 3.29·23-s + 2.56·24-s + 8.13·25-s + 5.40·26-s − 1.48·27-s + 4.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s + 1.62·5-s + 1.04·6-s + 1.81·7-s − 0.353·8-s + 1.19·9-s − 1.14·10-s + 0.145·11-s − 0.740·12-s − 1.49·13-s − 1.28·14-s − 2.40·15-s + 0.250·16-s − 1.36·17-s − 0.843·18-s − 0.229·19-s + 0.810·20-s − 2.69·21-s − 0.103·22-s − 0.686·23-s + 0.523·24-s + 1.62·25-s + 1.06·26-s − 0.286·27-s + 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 - 4.81T + 7T^{2} \) |
| 11 | \( 1 - 0.483T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 + 5.64T + 17T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 - 7.39T + 29T^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 37 | \( 1 + 4.09T + 37T^{2} \) |
| 41 | \( 1 - 3.25T + 41T^{2} \) |
| 43 | \( 1 + 9.51T + 43T^{2} \) |
| 47 | \( 1 + 1.74T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 - 0.239T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 0.0139T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 4.43T + 79T^{2} \) |
| 83 | \( 1 + 3.63T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 7.06T + 97T^{2} \) |
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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
−7.36098920286754513712057317125, −6.63592519228342542156191615825, −6.22513589817332268651084048176, −5.25702470170169728327703929529, −5.04977807445576186882689880914, −4.38637677099137181732397645065, −2.52717510352741671799054984578, −1.93134762340573115602053783819, −1.29913586826458149623850348836, 0,
1.29913586826458149623850348836, 1.93134762340573115602053783819, 2.52717510352741671799054984578, 4.38637677099137181732397645065, 5.04977807445576186882689880914, 5.25702470170169728327703929529, 6.22513589817332268651084048176, 6.63592519228342542156191615825, 7.36098920286754513712057317125
