L(s) = 1 | + (0.213 − 1.39i)2-s + (−1.90 − 0.595i)4-s + (1.90 + 1.17i)5-s + (2.16 + 2.16i)7-s + (−1.23 + 2.54i)8-s + (2.04 − 2.40i)10-s + (−3.61 − 4.97i)11-s + (0.749 + 4.72i)13-s + (3.48 − 2.56i)14-s + (3.29 + 2.27i)16-s + (3.51 + 1.79i)17-s + (0.678 + 2.08i)19-s + (−2.93 − 3.37i)20-s + (−7.73 + 3.99i)22-s + (−0.374 + 2.36i)23-s + ⋯ |
L(s) = 1 | + (0.150 − 0.988i)2-s + (−0.954 − 0.297i)4-s + (0.850 + 0.525i)5-s + (0.817 + 0.817i)7-s + (−0.438 + 0.898i)8-s + (0.647 − 0.762i)10-s + (−1.09 − 1.50i)11-s + (0.207 + 1.31i)13-s + (0.931 − 0.685i)14-s + (0.822 + 0.568i)16-s + (0.852 + 0.434i)17-s + (0.155 + 0.479i)19-s + (−0.655 − 0.754i)20-s + (−1.64 + 0.852i)22-s + (−0.0781 + 0.493i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81015 - 0.311325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81015 - 0.311325i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.213 + 1.39i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.90 - 1.17i)T \) |
good | 7 | \( 1 + (-2.16 - 2.16i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.61 + 4.97i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.749 - 4.72i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.51 - 1.79i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.678 - 2.08i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.374 - 2.36i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (1.48 + 0.480i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-8.29 + 2.69i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 0.644i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.32 - 3.86i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (8.72 - 8.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.47 + 1.77i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.54 + 2.82i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (7.51 + 5.46i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.715 - 0.519i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.52 + 2.98i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (5.96 + 1.93i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.906 - 0.143i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.98 + 9.17i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.49 - 3.31i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (0.440 + 0.606i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (7.49 + 14.7i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.11802103674572688480552728599, −9.453670277433350561674761333761, −8.500509693527912398790480486167, −7.910909086021980594454525977230, −6.15217810417298522146555442964, −5.69373328375882962839800496002, −4.73830314057303887329945314908, −3.36989837021267752598167538843, −2.48709498532910147672588618672, −1.45883775908046964418610106721,
0.960439501459766255171930136021, 2.69212998620400082975752705063, 4.31951277903398438477799639001, 5.05016288608265463282294478621, 5.58551003484206922869247025839, 6.84936823697857497011602515615, 7.68180491952461002725909065479, 8.154328875528721060265077035300, 9.246210698966387276688722642291, 10.22489597104581188439777942464