L(s) = 1 | + (−2.42 + 0.649i)2-s + (−1.28 − 1.16i)3-s + (3.72 − 2.14i)4-s + (3.58 − 0.959i)5-s + (3.86 + 1.98i)6-s + (1.55 − 2.14i)7-s + (−4.07 + 4.07i)8-s + (0.301 + 2.98i)9-s + (−8.06 + 4.65i)10-s + (−2.52 − 0.677i)11-s + (−7.27 − 1.56i)12-s + (3.59 − 0.242i)13-s + (−2.37 + 6.20i)14-s + (−5.71 − 2.92i)15-s + (2.93 − 5.08i)16-s + (−0.358 − 0.621i)17-s + ⋯ |
L(s) = 1 | + (−1.71 + 0.459i)2-s + (−0.741 − 0.670i)3-s + (1.86 − 1.07i)4-s + (1.60 − 0.429i)5-s + (1.57 + 0.808i)6-s + (0.586 − 0.809i)7-s + (−1.44 + 1.44i)8-s + (0.100 + 0.994i)9-s + (−2.54 + 1.47i)10-s + (−0.761 − 0.204i)11-s + (−2.10 − 0.450i)12-s + (0.997 − 0.0671i)13-s + (−0.634 + 1.65i)14-s + (−1.47 − 0.755i)15-s + (0.733 − 1.27i)16-s + (−0.0870 − 0.150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.561842 - 0.284601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.561842 - 0.284601i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.28 + 1.16i)T \) |
| 7 | \( 1 + (-1.55 + 2.14i)T \) |
| 13 | \( 1 + (-3.59 + 0.242i)T \) |
good | 2 | \( 1 + (2.42 - 0.649i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-3.58 + 0.959i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (2.52 + 0.677i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.358 + 0.621i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.19 - 4.46i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.80 + 4.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.19iT - 29T^{2} \) |
| 31 | \( 1 + (3.93 + 1.05i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (5.34 - 1.43i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.34 - 3.34i)T + 41iT^{2} \) |
| 43 | \( 1 + 4.83iT - 43T^{2} \) |
| 47 | \( 1 + (0.235 + 0.879i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.40 + 1.38i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.51 + 2.28i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-5.31 + 9.21i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.15 + 0.844i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-7.32 - 7.32i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.53 - 5.73i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.74 - 3.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.93 + 1.93i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.74 + 10.2i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (13.6 - 13.6i)T - 97iT^{2} \) |
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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
−11.19171283923357722561571427728, −10.56851494781589896567055060765, −9.971624624193778769047225058338, −8.754850512381663491465107327456, −7.992375773441832515227827295290, −6.93616213559414101013010983250, −6.07054686551712435884560347512, −5.21677991248586977276135968161, −1.99101074722664212411649223369, −1.02374296834741558106126508374,
1.59493508446747760561766733364, 2.87123756626455486661186063325, 5.24432458687195771014761192973, 6.11768388880698619990680273583, 7.27392863062268436911875677008, 8.797106644060159566605624494033, 9.277925252701226716756995867620, 10.12583321948107880761802042561, 10.90862175465518641530689668015, 11.34611602423473075493503817143