| L(s) = 1 | + 2-s − 3-s + 4-s − 0.487·5-s − 6-s + 4.75·7-s + 8-s + 9-s − 0.487·10-s − 5.66·11-s − 12-s + 4.03·13-s + 4.75·14-s + 0.487·15-s + 16-s − 4.44·17-s + 18-s + 19-s − 0.487·20-s − 4.75·21-s − 5.66·22-s − 5.49·23-s − 24-s − 4.76·25-s + 4.03·26-s − 27-s + 4.75·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.217·5-s − 0.408·6-s + 1.79·7-s + 0.353·8-s + 0.333·9-s − 0.154·10-s − 1.70·11-s − 0.288·12-s + 1.11·13-s + 1.26·14-s + 0.125·15-s + 0.250·16-s − 1.07·17-s + 0.235·18-s + 0.229·19-s − 0.108·20-s − 1.03·21-s − 1.20·22-s − 1.14·23-s − 0.204·24-s − 0.952·25-s + 0.790·26-s − 0.192·27-s + 0.897·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 0.487T + 5T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 11 | \( 1 + 5.66T + 11T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 - 0.0939T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 6.52T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 0.302T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 0.0301T + 79T^{2} \) |
| 83 | \( 1 + 5.61T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
| 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
−7.57540710112519962570267386136, −7.12938235480384853888657060254, −5.94334749943294982997700038051, −5.52816257591455334032734774555, −4.86186390597963263918729245159, −4.27093190879404211687913401727, −3.42490942792825554921993438559, −2.12719828227820341572428166625, −1.64039392811028791958701503413, 0,
1.64039392811028791958701503413, 2.12719828227820341572428166625, 3.42490942792825554921993438559, 4.27093190879404211687913401727, 4.86186390597963263918729245159, 5.52816257591455334032734774555, 5.94334749943294982997700038051, 7.12938235480384853888657060254, 7.57540710112519962570267386136
