L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (0.500 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s + (−0.939 − 0.342i)22-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (0.499 + 0.866i)27-s + (0.939 − 0.342i)32-s + (−0.173 − 0.984i)33-s + (−0.347 + 1.96i)34-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (0.500 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s + (−0.939 − 0.342i)22-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (0.499 + 0.866i)27-s + (0.939 − 0.342i)32-s + (−0.173 − 0.984i)33-s + (−0.347 + 1.96i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5903622885\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5903622885\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-1.87 + 0.684i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.020648447445980875506651196114, −8.033274411837153025327852445110, −7.36671322863315051582900915883, −6.78885866963330162267665106654, −6.04016531276623448870764209097, −5.34519355734521744266417620864, −4.71573727638766214188246433440, −3.30487519179585972210789711705, −1.88754771767008998070881159250, −0.806607480943845926828533750502,
0.869734020773896771011876679620, 2.08805506983884222471264065174, 3.40664048697879073138435065179, 3.93622838396439939386872983214, 5.13215657771879929605651373264, 5.94045596402613261235641502634, 6.55207868240703066513607399430, 7.73181212107208513787750763001, 8.231872127757781998845821792937, 8.996541679778081359142193705610