| L(s) = 1 | + (−0.909 + 0.415i)2-s + (−0.768 − 2.61i)3-s + (0.654 − 0.755i)4-s + (1.33 − 1.79i)5-s + (1.78 + 2.06i)6-s + (1.27 − 0.182i)7-s + (−0.281 + 0.959i)8-s + (−3.73 + 2.40i)9-s + (−0.465 + 2.18i)10-s + (1.03 − 2.27i)11-s + (−2.48 − 1.13i)12-s + (−1.02 − 0.146i)13-s + (−1.07 + 0.693i)14-s + (−5.72 − 2.10i)15-s + (−0.142 − 0.989i)16-s + (−3.98 + 3.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 + 0.293i)2-s + (−0.443 − 1.51i)3-s + (0.327 − 0.377i)4-s + (0.595 − 0.803i)5-s + (0.729 + 0.841i)6-s + (0.480 − 0.0690i)7-s + (−0.0996 + 0.339i)8-s + (−1.24 + 0.800i)9-s + (−0.147 + 0.691i)10-s + (0.313 − 0.686i)11-s + (−0.716 − 0.327i)12-s + (−0.283 − 0.0406i)13-s + (−0.288 + 0.185i)14-s + (−1.47 − 0.543i)15-s + (−0.0355 − 0.247i)16-s + (−0.965 + 0.837i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.433654 - 0.718442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.433654 - 0.718442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 - 0.415i)T \) |
| 5 | \( 1 + (-1.33 + 1.79i)T \) |
| 23 | \( 1 + (2.14 + 4.28i)T \) |
| good | 3 | \( 1 + (0.768 + 2.61i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-1.27 + 0.182i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.03 + 2.27i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (1.02 + 0.146i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (3.98 - 3.45i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.540 + 0.623i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 1.28i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-7.72 - 2.26i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-5.04 - 7.85i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (3.34 + 2.15i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.74 - 9.34i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 10.1iT - 47T^{2} \) |
| 53 | \( 1 + (-3.94 + 0.567i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.11 + 7.73i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (3.40 + 0.998i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-13.0 + 5.97i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.65 - 5.82i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-12.6 - 10.9i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.01 + 7.07i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.41 - 3.75i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-6.62 + 1.94i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-8.57 + 13.3i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−11.93818817312254811775323283161, −11.08679420110697965876972744792, −9.864556891015967885739055076304, −8.464179167237016757234538906050, −8.167034984122521758059007109681, −6.67125512776376410928495685396, −6.16846902870444345922369878972, −4.86500158241408579700708403418, −2.15142637002093553238042757190, −0.935803266287973397676343931005,
2.41420811780754217030098312964, 3.96044962960672350311802633457, 5.10604684756572346310881482470, 6.39088011467265838932544624579, 7.63541000957685907953998662159, 9.198340372531909704943786928088, 9.676218301604549765496367445557, 10.50044349108659375865357633048, 11.25105620052881141687911977779, 11.95952355317509170537541358309
