L(s) = 1 | + (1.65 − 2.87i)3-s + (1.65 + 2.87i)7-s + (−4 − 6.92i)9-s + (1.65 − 2.87i)11-s + (−1 − 3.46i)13-s + (1.5 + 2.59i)17-s + (1.65 + 2.87i)19-s + 11·21-s + (−1.65 + 2.87i)23-s − 5·25-s − 16.5·27-s + (2.5 − 4.33i)29-s + (−5.5 − 9.52i)33-s + (−4.5 + 7.79i)37-s + (−11.6 − 2.87i)39-s + ⋯ |
L(s) = 1 | + (0.957 − 1.65i)3-s + (0.626 + 1.08i)7-s + (−1.33 − 2.30i)9-s + (0.500 − 0.866i)11-s + (−0.277 − 0.960i)13-s + (0.363 + 0.630i)17-s + (0.380 + 0.658i)19-s + 2.40·21-s + (−0.345 + 0.598i)23-s − 25-s − 3.19·27-s + (0.464 − 0.804i)29-s + (−0.957 − 1.65i)33-s + (−0.739 + 1.28i)37-s + (−1.85 − 0.459i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36400 - 1.34662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36400 - 1.34662i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 3 | \( 1 + (-1.65 + 2.87i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + (-1.65 - 2.87i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.65 + 2.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.65 - 2.87i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.65 - 2.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.97 - 8.61i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + (-1.65 - 2.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.97 + 8.61i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.97 - 8.61i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 6.63T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.43994153228692479353828043939, −9.834054597293179805332052482999, −8.760104854956895070987330381274, −8.137168737027138719099948848284, −7.60955100485619283118037922508, −6.20630909665338915881201577156, −5.66840111213850985847655353908, −3.53363272033310291419500223737, −2.48349446992287228183830790898, −1.31218675849413734944107603764,
2.21698153858537655787529738817, 3.73067982323631693501141951886, 4.38431073096514722446644601749, 5.15945733075600577077676271597, 7.05029437252841768071911368005, 7.87273121172963451066923323974, 9.035589203665678882415639145963, 9.555010468223417486345730071864, 10.41146494491583582180983605036, 11.08393222630473618117997142836