| L(s) = 1 | + 0.149·2-s + 1.36·3-s − 1.97·4-s − 3.17·5-s + 0.203·6-s + 3.47·7-s − 0.594·8-s − 1.14·9-s − 0.474·10-s + 0.0807·11-s − 2.69·12-s + 5.65·13-s + 0.520·14-s − 4.31·15-s + 3.86·16-s − 5.35·17-s − 0.171·18-s − 3.63·19-s + 6.27·20-s + 4.73·21-s + 0.0120·22-s − 1.35·23-s − 0.809·24-s + 5.05·25-s + 0.846·26-s − 5.64·27-s − 6.87·28-s + ⋯ |
| L(s) = 1 | + 0.105·2-s + 0.786·3-s − 0.988·4-s − 1.41·5-s + 0.0831·6-s + 1.31·7-s − 0.210·8-s − 0.381·9-s − 0.149·10-s + 0.0243·11-s − 0.777·12-s + 1.56·13-s + 0.138·14-s − 1.11·15-s + 0.966·16-s − 1.29·17-s − 0.0403·18-s − 0.834·19-s + 1.40·20-s + 1.03·21-s + 0.00257·22-s − 0.281·23-s − 0.165·24-s + 1.01·25-s + 0.165·26-s − 1.08·27-s − 1.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 0.149T + 2T^{2} \) |
| 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 - 3.47T + 7T^{2} \) |
| 11 | \( 1 - 0.0807T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 + 0.781T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 3.42T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 + 6.49T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 5.99T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.567205468960747378192469090730, −8.306008036220846959657797977525, −7.74505560704889668486296567538, −6.52313849222701366197639290731, −5.39103078284592584303213058952, −4.35013672985756907663419228804, −4.03867491877328293300163662859, −3.09345470421011618673511901808, −1.62009626571915922564449172401, 0,
1.62009626571915922564449172401, 3.09345470421011618673511901808, 4.03867491877328293300163662859, 4.35013672985756907663419228804, 5.39103078284592584303213058952, 6.52313849222701366197639290731, 7.74505560704889668486296567538, 8.306008036220846959657797977525, 8.567205468960747378192469090730
