L(s) = 1 | − 2.67·2-s − 3-s + 5.15·4-s + 2.67·6-s + 2.28·7-s − 8.44·8-s + 9-s − 0.130·11-s − 5.15·12-s + 13-s − 6.11·14-s + 12.2·16-s − 7.96·17-s − 2.67·18-s − 5.11·19-s − 2.28·21-s + 0.350·22-s + 5.50·23-s + 8.44·24-s − 2.67·26-s − 27-s + 11.7·28-s − 3·29-s − 7.79·31-s − 15.9·32-s + 0.130·33-s + 21.3·34-s + ⋯ |
L(s) = 1 | − 1.89·2-s − 0.577·3-s + 2.57·4-s + 1.09·6-s + 0.864·7-s − 2.98·8-s + 0.333·9-s − 0.0394·11-s − 1.48·12-s + 0.277·13-s − 1.63·14-s + 3.06·16-s − 1.93·17-s − 0.630·18-s − 1.17·19-s − 0.499·21-s + 0.0746·22-s + 1.14·23-s + 1.72·24-s − 0.524·26-s − 0.192·27-s + 2.22·28-s − 0.557·29-s − 1.39·31-s − 2.81·32-s + 0.0227·33-s + 3.65·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 + 0.130T + 11T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 + 2.67T + 47T^{2} \) |
| 53 | \( 1 - 8.50T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 5.18T + 61T^{2} \) |
| 67 | \( 1 - 6.05T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 0.932T + 73T^{2} \) |
| 79 | \( 1 - 8.85T + 79T^{2} \) |
| 83 | \( 1 + 6.80T + 83T^{2} \) |
| 89 | \( 1 + 8.12T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−9.342321535029925507437342007434, −8.883807448393710359142409598196, −8.113411267887335624816355162480, −7.16666235438336442874409726232, −6.60595514649016314680584886507, −5.55820437378876826738013433815, −4.24844369552910865924735403528, −2.44270107509509900594424383456, −1.50918111094437902559098330188, 0,
1.50918111094437902559098330188, 2.44270107509509900594424383456, 4.24844369552910865924735403528, 5.55820437378876826738013433815, 6.60595514649016314680584886507, 7.16666235438336442874409726232, 8.113411267887335624816355162480, 8.883807448393710359142409598196, 9.342321535029925507437342007434