L(s) = 1 | − 0.586·2-s − 2.44·3-s − 1.65·4-s + 1.13·5-s + 1.43·6-s + 1.52·7-s + 2.14·8-s + 2.96·9-s − 0.664·10-s − 1.04·11-s + 4.04·12-s − 5.76·13-s − 0.896·14-s − 2.76·15-s + 2.05·16-s + 3.08·17-s − 1.74·18-s + 1.56·19-s − 1.87·20-s − 3.73·21-s + 0.613·22-s − 3.70·23-s − 5.24·24-s − 3.71·25-s + 3.38·26-s + 0.0856·27-s − 2.52·28-s + ⋯ |
L(s) = 1 | − 0.415·2-s − 1.41·3-s − 0.827·4-s + 0.506·5-s + 0.585·6-s + 0.577·7-s + 0.758·8-s + 0.988·9-s − 0.210·10-s − 0.315·11-s + 1.16·12-s − 1.59·13-s − 0.239·14-s − 0.713·15-s + 0.512·16-s + 0.747·17-s − 0.410·18-s + 0.358·19-s − 0.419·20-s − 0.814·21-s + 0.130·22-s − 0.771·23-s − 1.06·24-s − 0.743·25-s + 0.663·26-s + 0.0164·27-s − 0.478·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4900229160\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4900229160\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 8011 | \( 1+O(T) \) |
good | 2 | \( 1 + 0.586T + 2T^{2} \) |
| 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 + 5.76T + 13T^{2} \) |
| 17 | \( 1 - 3.08T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 + 9.41T + 43T^{2} \) |
| 47 | \( 1 + 0.750T + 47T^{2} \) |
| 53 | \( 1 - 7.83T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 - 6.53T + 61T^{2} \) |
| 67 | \( 1 - 9.84T + 67T^{2} \) |
| 71 | \( 1 - 2.71T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 - 7.29T + 79T^{2} \) |
| 83 | \( 1 + 3.32T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 1.76T + 97T^{2} \) |
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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
−7.75245200215528930490822228199, −7.29854535231892309223837163817, −6.26894177172555092431853565513, −5.60410577380394665318352119020, −5.09370482210285799979047358193, −4.67200934549618280861791344194, −3.74439746102490455417571904282, −2.42353864452172995967816504395, −1.44692861760281384737663858548, −0.42397518229256327952695155918,
0.42397518229256327952695155918, 1.44692861760281384737663858548, 2.42353864452172995967816504395, 3.74439746102490455417571904282, 4.67200934549618280861791344194, 5.09370482210285799979047358193, 5.60410577380394665318352119020, 6.26894177172555092431853565513, 7.29854535231892309223837163817, 7.75245200215528930490822228199