L(s) = 1 | − 8.82i·7-s − 9·9-s + 17.3·11-s − 33.9i·17-s + 19·19-s + 30i·23-s − 31.1i·43-s + 11.5i·47-s − 28.8·49-s − 108.·61-s + 79.4i·63-s − 137. i·73-s − 153. i·77-s + 81·81-s − 90i·83-s + ⋯ |
L(s) = 1 | − 1.26i·7-s − 9-s + 1.57·11-s − 1.99i·17-s + 19-s + 1.30i·23-s − 0.725i·43-s + 0.246i·47-s − 0.589·49-s − 1.77·61-s + 1.26i·63-s − 1.87i·73-s − 1.99i·77-s + 81-s − 1.08i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.637992078\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.637992078\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
good | 3 | \( 1 + 9T^{2} \) |
| 7 | \( 1 + 8.82iT - 49T^{2} \) |
| 11 | \( 1 - 17.3T + 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 33.9iT - 289T^{2} \) |
| 23 | \( 1 - 30iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 31.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 90iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−9.168255915948702844641858091679, −7.76351187778918195207792426526, −7.27926352759772441867305707083, −6.51569113508057661306949954467, −5.53099272534833963374954387871, −4.66315781078995263602336963644, −3.67019378559304485186281163144, −2.99380175250934575305332948287, −1.42747722735286990211846970394, −0.45288250904987640902847881987,
1.30373535583848900119308089167, 2.39830462726372269722621045547, 3.38153463634724409254475836073, 4.30594547665405074871127309417, 5.47112434530816358693177120893, 6.10487747646180884592202170228, 6.63946738744601313995562183086, 7.977540450017471918655036957307, 8.646589718687189536878003986663, 9.052333284661403527719205105885