| L(s) = 1 | − 0.724·2-s − 1.47·4-s + 1.17·5-s − 4.09·7-s + 2.51·8-s − 0.853·10-s + 3.08·11-s + 2.42·13-s + 2.96·14-s + 1.12·16-s − 0.237·17-s − 2.02·19-s − 1.73·20-s − 2.23·22-s + 23-s − 3.61·25-s − 1.75·26-s + 6.03·28-s + 29-s + 0.00576·31-s − 5.85·32-s + 0.171·34-s − 4.82·35-s + 1.67·37-s + 1.46·38-s + 2.96·40-s + 7.50·41-s + ⋯ |
| L(s) = 1 | − 0.512·2-s − 0.737·4-s + 0.527·5-s − 1.54·7-s + 0.889·8-s − 0.269·10-s + 0.930·11-s + 0.673·13-s + 0.791·14-s + 0.282·16-s − 0.0575·17-s − 0.464·19-s − 0.388·20-s − 0.476·22-s + 0.208·23-s − 0.722·25-s − 0.344·26-s + 1.14·28-s + 0.185·29-s + 0.00103·31-s − 1.03·32-s + 0.0294·34-s − 0.815·35-s + 0.275·37-s + 0.237·38-s + 0.469·40-s + 1.17·41-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
| \(L(1)\) |
\(\approx\) |
\(1.006692027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006692027\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.724T + 2T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 - 3.08T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 + 0.237T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 31 | \( 1 - 0.00576T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 - 7.50T + 41T^{2} \) |
| 43 | \( 1 - 6.04T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 0.167T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 4.61T + 83T^{2} \) |
| 89 | \( 1 - 2.09T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−8.213701277921240020868428983150, −7.44083179490894918095006509083, −6.46435700068084393761162105347, −6.20166894701844651301257438421, −5.32486432430431372537666017392, −4.21942848381513029898813040391, −3.76167507615012376020882287205, −2.81736257127310477799388988120, −1.62207505947745978457795487143, −0.59298207598537322162049364220,
0.59298207598537322162049364220, 1.62207505947745978457795487143, 2.81736257127310477799388988120, 3.76167507615012376020882287205, 4.21942848381513029898813040391, 5.32486432430431372537666017392, 6.20166894701844651301257438421, 6.46435700068084393761162105347, 7.44083179490894918095006509083, 8.213701277921240020868428983150
