L(s) = 1 | + (1 + i)2-s + (4.03 − 4.03i)3-s + 2i·4-s + (−1.56 − 4.74i)5-s + 8.07·6-s + (6.17 + 6.17i)7-s + (−2 + 2i)8-s − 23.5i·9-s + (3.17 − 6.31i)10-s − 0.724·11-s + (8.07 + 8.07i)12-s + (−8.24 + 8.24i)13-s + 12.3i·14-s + (−25.4 − 12.8i)15-s − 4·16-s + (12.1 + 12.1i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (1.34 − 1.34i)3-s + 0.5i·4-s + (−0.313 − 0.949i)5-s + 1.34·6-s + (0.881 + 0.881i)7-s + (−0.250 + 0.250i)8-s − 2.61i·9-s + (0.317 − 0.631i)10-s − 0.0658·11-s + (0.672 + 0.672i)12-s + (−0.634 + 0.634i)13-s + 0.881i·14-s + (−1.69 − 0.855i)15-s − 0.250·16-s + (0.714 + 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.82579 - 1.02861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.82579 - 1.02861i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 + (1.56 + 4.74i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
good | 3 | \( 1 + (-4.03 + 4.03i)T - 9iT^{2} \) |
| 7 | \( 1 + (-6.17 - 6.17i)T + 49iT^{2} \) |
| 11 | \( 1 + 0.724T + 121T^{2} \) |
| 13 | \( 1 + (8.24 - 8.24i)T - 169iT^{2} \) |
| 17 | \( 1 + (-12.1 - 12.1i)T + 289iT^{2} \) |
| 19 | \( 1 + 25.7iT - 361T^{2} \) |
| 29 | \( 1 - 44.6iT - 841T^{2} \) |
| 31 | \( 1 + 21.4T + 961T^{2} \) |
| 37 | \( 1 + (6.02 + 6.02i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 62.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (22.3 - 22.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-59.0 - 59.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (23.4 - 23.4i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 65.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4.47T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-3.14 - 3.14i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 116.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (23.6 - 23.6i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 35.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-24.3 + 24.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 95.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (33.7 + 33.7i)T + 9.40e3iT^{2} \) |
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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
−12.44360717723356250937903116018, −11.46359234354343665451892477343, −9.092220919183905361919650165470, −8.818492596732877954902937596343, −7.80893825061939735371983440026, −7.15525840122803595043579465412, −5.73133036555076389798857908429, −4.42605646743591417675882055005, −2.83183265262319489968277393312, −1.54225477878186196977250770547,
2.34673558451811094466183206840, 3.48868843404392011293010803947, 4.20679958188763859802573727609, 5.38067752757367233921311459715, 7.51745489873586970778621414414, 8.024714741562458141609964692900, 9.552486732519460294308656697743, 10.27612837129318547285425994413, 10.78990178341246063264474435374, 11.86553224230942747055608547731