L(s) = 1 | + (−0.970 − 1.68i)2-s + 3·3-s + (2.11 − 3.66i)4-s + 11.3·5-s + (−2.91 − 5.04i)6-s + (14.7 − 25.6i)7-s − 23.7·8-s + 9·9-s + (−11.0 − 19.1i)10-s + (−1.65 + 2.86i)11-s + (6.34 − 10.9i)12-s + (3.34 + 5.79i)13-s − 57.4·14-s + 34.1·15-s + (6.12 + 10.6i)16-s + (29.4 + 50.9i)17-s + ⋯ |
L(s) = 1 | + (−0.343 − 0.594i)2-s + 0.577·3-s + (0.264 − 0.457i)4-s + 1.01·5-s + (−0.198 − 0.343i)6-s + (0.798 − 1.38i)7-s − 1.04·8-s + 0.333·9-s + (−0.349 − 0.604i)10-s + (−0.0453 + 0.0785i)11-s + (0.152 − 0.264i)12-s + (0.0713 + 0.123i)13-s − 1.09·14-s + 0.587·15-s + (0.0957 + 0.165i)16-s + (0.419 + 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.46439 - 1.83210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46439 - 1.83210i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 + (-411. + 362. i)T \) |
good | 2 | \( 1 + (0.970 + 1.68i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 11.3T + 125T^{2} \) |
| 7 | \( 1 + (-14.7 + 25.6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (1.65 - 2.86i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-3.34 - 5.79i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-29.4 - 50.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-8.98 - 15.5i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-12.7 - 22.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-61.4 + 106. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (67.3 - 116. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (172. + 298. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (57.5 - 99.6i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + 45.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + (19.1 - 33.1i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 160.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 383.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (26.2 + 45.5i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 71 | \( 1 + (401. - 695. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (166. + 288. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (92.7 - 160. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-721. - 1.24e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 198.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-394. - 683. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.43294024485033023779136363975, −10.44756096547444371584146453547, −10.04539936826516699309373604463, −8.973534906530219716700139117413, −7.75928146095955364191510698784, −6.58852970447247423382008201355, −5.32413693906883731866365951910, −3.76414643373039534507601896809, −2.10962231447554325560677432055, −1.16070099881638034372618849761,
2.00436436811127759770000496875, 3.03473121680262054237365063675, 5.14041224014627875619883278426, 6.08867155614700616330833222031, 7.31327738063616567806738063262, 8.432361521169206909588127238709, 8.964837407199312254908707695343, 9.965500942170592864765633348362, 11.47731837883507087011394429502, 12.24591205419167277469693819465