| L(s) = 1 | − 2.26·2-s − 1.05·3-s + 3.12·4-s + 5-s + 2.39·6-s + 1.78·7-s − 2.55·8-s − 1.88·9-s − 2.26·10-s − 3.65·11-s − 3.30·12-s − 3.53·13-s − 4.05·14-s − 1.05·15-s − 0.469·16-s − 5.83·17-s + 4.26·18-s − 4.81·19-s + 3.12·20-s − 1.89·21-s + 8.26·22-s + 2.44·23-s + 2.69·24-s + 25-s + 7.99·26-s + 5.15·27-s + 5.59·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 0.609·3-s + 1.56·4-s + 0.447·5-s + 0.976·6-s + 0.676·7-s − 0.903·8-s − 0.628·9-s − 0.716·10-s − 1.10·11-s − 0.953·12-s − 0.979·13-s − 1.08·14-s − 0.272·15-s − 0.117·16-s − 1.41·17-s + 1.00·18-s − 1.10·19-s + 0.699·20-s − 0.412·21-s + 1.76·22-s + 0.510·23-s + 0.550·24-s + 0.200·25-s + 1.56·26-s + 0.992·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3679833278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3679833278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 + 1.05T + 3T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 0.874T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 + 5.15T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 - 2.78T + 61T^{2} \) |
| 67 | \( 1 - 0.593T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 0.882T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.927962561894613172884838679642, −8.631251003401389104652826170060, −7.73540303313421175529107280643, −7.02373897744657974160445950676, −6.21914915940731721153991089857, −5.23605246501513963627128133051, −4.53919153662533928496746134321, −2.58878755549338813853283542097, −2.06213633818677117674938584687, −0.50089398036772392058425026189,
0.50089398036772392058425026189, 2.06213633818677117674938584687, 2.58878755549338813853283542097, 4.53919153662533928496746134321, 5.23605246501513963627128133051, 6.21914915940731721153991089857, 7.02373897744657974160445950676, 7.73540303313421175529107280643, 8.631251003401389104652826170060, 8.927962561894613172884838679642
