L(s) = 1 | − 2.45·2-s + 2.98·3-s + 4.04·4-s − 1.52·5-s − 7.34·6-s + 5.18·7-s − 5.01·8-s + 5.92·9-s + 3.75·10-s − 3.38·11-s + 12.0·12-s + 13-s − 12.7·14-s − 4.56·15-s + 4.24·16-s − 6.38·17-s − 14.5·18-s − 0.342·19-s − 6.17·20-s + 15.4·21-s + 8.33·22-s − 2.04·23-s − 14.9·24-s − 2.66·25-s − 2.45·26-s + 8.72·27-s + 20.9·28-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 1.72·3-s + 2.02·4-s − 0.683·5-s − 2.99·6-s + 1.95·7-s − 1.77·8-s + 1.97·9-s + 1.18·10-s − 1.02·11-s + 3.48·12-s + 0.277·13-s − 3.40·14-s − 1.17·15-s + 1.06·16-s − 1.54·17-s − 3.43·18-s − 0.0785·19-s − 1.37·20-s + 3.37·21-s + 1.77·22-s − 0.425·23-s − 3.05·24-s − 0.533·25-s − 0.481·26-s + 1.67·27-s + 3.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 - 5.18T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 + 0.342T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.408T + 31T^{2} \) |
| 37 | \( 1 + 6.11T + 37T^{2} \) |
| 41 | \( 1 - 7.78T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 + 7.49T + 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 - 3.51T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.69341307684410177077536443726, −7.56591358866770320041335367142, −6.68569944326868509607165933973, −5.34599163142532319994472958180, −4.41377373646595331564183472905, −3.75686057599269068300232783278, −2.57533909226168964656650270512, −2.01594638448538686140630148878, −1.52435860824979313432587944173, 0,
1.52435860824979313432587944173, 2.01594638448538686140630148878, 2.57533909226168964656650270512, 3.75686057599269068300232783278, 4.41377373646595331564183472905, 5.34599163142532319994472958180, 6.68569944326868509607165933973, 7.56591358866770320041335367142, 7.69341307684410177077536443726