L(s) = 1 | − 2.70·2-s + 5.30·4-s − 2.79·5-s + 0.880·7-s − 8.93·8-s + 7.56·10-s + 5.67·11-s − 3.71·13-s − 2.38·14-s + 13.5·16-s − 4.80·17-s + 6.68·19-s − 14.8·20-s − 15.3·22-s − 23-s + 2.83·25-s + 10.0·26-s + 4.67·28-s + 29-s + 0.779·31-s − 18.7·32-s + 12.9·34-s − 2.46·35-s − 8.14·37-s − 18.0·38-s + 25.0·40-s − 6.06·41-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 2.65·4-s − 1.25·5-s + 0.332·7-s − 3.16·8-s + 2.39·10-s + 1.71·11-s − 1.03·13-s − 0.636·14-s + 3.38·16-s − 1.16·17-s + 1.53·19-s − 3.32·20-s − 3.27·22-s − 0.208·23-s + 0.567·25-s + 1.97·26-s + 0.883·28-s + 0.185·29-s + 0.139·31-s − 3.31·32-s + 2.22·34-s − 0.416·35-s − 1.33·37-s − 2.93·38-s + 3.95·40-s − 0.946·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 - 0.880T + 7T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 19 | \( 1 - 6.68T + 19T^{2} \) |
| 31 | \( 1 - 0.779T + 31T^{2} \) |
| 37 | \( 1 + 8.14T + 37T^{2} \) |
| 41 | \( 1 + 6.06T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 - 9.73T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 + 7.99T + 61T^{2} \) |
| 67 | \( 1 + 8.36T + 67T^{2} \) |
| 71 | \( 1 + 8.52T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 0.302T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 0.884T + 97T^{2} \) |
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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
−7.60621313912548003631727553892, −7.40409069019878618812812127147, −6.78797871212767481004682786004, −5.99094839906485783113193596339, −4.76065083486792800809061728639, −3.84390450242475773811154259203, −2.98785050087900601488583290731, −1.92542415422370635278339264144, −1.03515048346275653796417884093, 0,
1.03515048346275653796417884093, 1.92542415422370635278339264144, 2.98785050087900601488583290731, 3.84390450242475773811154259203, 4.76065083486792800809061728639, 5.99094839906485783113193596339, 6.78797871212767481004682786004, 7.40409069019878618812812127147, 7.60621313912548003631727553892