L(s) = 1 | + (0.906 + 0.422i)2-s + (−0.199 − 0.284i)3-s + (0.642 + 0.766i)4-s + (−0.0603 − 0.342i)6-s + (0.474 + 0.127i)7-s + (0.258 + 0.965i)8-s + (0.984 − 2.70i)9-s + (−2.42 + 4.19i)11-s + (0.0898 − 0.335i)12-s + (3.72 + 2.60i)13-s + (0.376 + 0.316i)14-s + (−0.173 + 0.984i)16-s + (2.78 − 5.97i)17-s + (2.03 − 2.03i)18-s + (3.35 + 2.78i)19-s + ⋯ |
L(s) = 1 | + (0.640 + 0.298i)2-s + (−0.115 − 0.164i)3-s + (0.321 + 0.383i)4-s + (−0.0246 − 0.139i)6-s + (0.179 + 0.0481i)7-s + (0.0915 + 0.341i)8-s + (0.328 − 0.901i)9-s + (−0.730 + 1.26i)11-s + (0.0259 − 0.0968i)12-s + (1.03 + 0.723i)13-s + (0.100 + 0.0844i)14-s + (−0.0434 + 0.246i)16-s + (0.676 − 1.44i)17-s + (0.479 − 0.479i)18-s + (0.768 + 0.639i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.27508 + 0.743380i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.27508 + 0.743380i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.906 - 0.422i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.35 - 2.78i)T \) |
good | 3 | \( 1 + (0.199 + 0.284i)T + (-1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-0.474 - 0.127i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (2.42 - 4.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.72 - 2.60i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.78 + 5.97i)T + (-10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (-1.10 - 0.0964i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (-5.74 - 2.08i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (7.25 - 4.18i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.26 - 7.26i)T + 37iT^{2} \) |
| 41 | \( 1 + (-5.95 - 1.05i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.785 + 8.97i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-4.53 + 2.11i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.679 + 7.76i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-5.32 + 1.93i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (10.6 - 8.94i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.262 + 0.563i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (2.13 - 2.54i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.588 + 0.412i)T + (24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (1.35 - 7.66i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.41 - 8.99i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (0.290 + 1.64i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (14.1 + 6.58i)T + (62.3 + 74.3i)T^{2} \) |
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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
−10.02208161314711145937236001759, −9.376488556708282087219005832877, −8.312355638638410480524160019617, −7.24338119869917568130606787715, −6.87733479913349271434610147263, −5.71387492241277861491596322150, −4.91740586746882236511519891089, −3.93448575013997390898600057816, −2.88166697418828625034031945514, −1.39959719975679551056093362486,
1.12910289216830322859450526010, 2.65595593245490425794237700583, 3.60625320718864640455239256815, 4.62456496864328987137558109792, 5.73962715413119701955171807062, 6.00675132327732997193268747683, 7.66205287509826185010770049675, 8.051016319892292341853281852928, 9.198026796678995824450394504756, 10.32486697895818023064920605332