L(s) = 1 | − 2-s − 3-s + 4-s − 4.04·5-s + 6-s + 7-s − 8-s + 9-s + 4.04·10-s + 5.74·11-s − 12-s − 14-s + 4.04·15-s + 16-s + 1.40·17-s − 18-s − 6.26·19-s − 4.04·20-s − 21-s − 5.74·22-s + 1.86·23-s + 24-s + 11.3·25-s − 27-s + 28-s − 3.20·29-s − 4.04·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.81·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.28·10-s + 1.73·11-s − 0.288·12-s − 0.267·14-s + 1.04·15-s + 0.250·16-s + 0.340·17-s − 0.235·18-s − 1.43·19-s − 0.905·20-s − 0.218·21-s − 1.22·22-s + 0.388·23-s + 0.204·24-s + 2.27·25-s − 0.192·27-s + 0.188·28-s − 0.595·29-s − 0.739·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 4.04T + 5T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 6.26T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 + 8.63T + 31T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 8.27T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 7.50T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 9.20T + 67T^{2} \) |
| 71 | \( 1 - 16.8T + 71T^{2} \) |
| 73 | \( 1 - 0.317T + 73T^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 + 7.10T + 83T^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57749115472305155629408709192, −6.98236423818887164975589258544, −6.51832245713235615884307887774, −5.53498866978288669005766059712, −4.54198332281166913382168497575, −3.93384977244362960011396599043, −3.42003462255525348509387762446, −1.95865328028201742720106199816, −0.990672632756559878822397403482, 0,
0.990672632756559878822397403482, 1.95865328028201742720106199816, 3.42003462255525348509387762446, 3.93384977244362960011396599043, 4.54198332281166913382168497575, 5.53498866978288669005766059712, 6.51832245713235615884307887774, 6.98236423818887164975589258544, 7.57749115472305155629408709192