L(s) = 1 | + (−1 + 1.73i)3-s + 5-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s + (2.5 + 2.59i)13-s + (−1 + 1.73i)15-s + (−1.5 − 2.59i)17-s + (1 + 1.73i)19-s + (−1 + 1.73i)23-s − 4·25-s − 4.00·27-s + (−2.5 + 4.33i)29-s + 2·31-s + (−3.99 − 6.92i)33-s + (−2.5 + 4.33i)37-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.999i)3-s + 0.447·5-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s + (0.693 + 0.720i)13-s + (−0.258 + 0.447i)15-s + (−0.363 − 0.630i)17-s + (0.229 + 0.397i)19-s + (−0.208 + 0.361i)23-s − 0.800·25-s − 0.769·27-s + (−0.464 + 0.804i)29-s + 0.359·31-s + (−0.696 − 1.20i)33-s + (−0.410 + 0.711i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.542525 + 0.912455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.542525 + 0.912455i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (-2.5 - 2.59i)T \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 13T + 53T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.36303698360904672155054736461, −10.49968898232328623053280887750, −9.818620437023757232388838948827, −9.133719478554666529755097779734, −7.80572251307510414354091038004, −6.71005873146354384170307873527, −5.55272950821372477571782559729, −4.78933096749039358130717592127, −3.76521666680915787686574047205, −2.01982745446997270784132451189,
0.74079073947940448674303916649, 2.31192757469160141331754803335, 3.85092699556815660406124571196, 5.67612001872800744029860585719, 5.93636381038984198334203286701, 7.11200806205607986272802979645, 8.057165138715353796603730874785, 8.943373161667052862379020486968, 10.24188008281258316411059250208, 10.98322650556110006452846065851