| L(s) = 1 | − 1.41·2-s + 3.34i·3-s + 2.00·4-s − 4.72i·6-s + 1.58·7-s − 2.82·8-s − 2.17·9-s + 2.76i·11-s + 6.68i·12-s − 14.7i·13-s − 2.24·14-s + 4.00·16-s − 17.5i·17-s + 3.07·18-s − 12.2·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.11i·3-s + 0.500·4-s − 0.787i·6-s + 0.226·7-s − 0.353·8-s − 0.241·9-s + 0.251i·11-s + 0.557i·12-s − 1.13i·13-s − 0.160·14-s + 0.250·16-s − 1.03i·17-s + 0.170·18-s − 0.642·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.076308391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076308391\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (-21.5 - 22.2i)T \) |
| good | 3 | \( 1 - 3.34iT - 9T^{2} \) |
| 7 | \( 1 - 1.58T + 49T^{2} \) |
| 11 | \( 1 - 2.76iT - 121T^{2} \) |
| 13 | \( 1 + 14.7iT - 169T^{2} \) |
| 17 | \( 1 + 17.5iT - 289T^{2} \) |
| 19 | \( 1 + 12.2T + 361T^{2} \) |
| 23 | \( 1 - 22.2iT - 529T^{2} \) |
| 29 | \( 1 + 14.7iT - 841T^{2} \) |
| 37 | \( 1 + 8.30iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 5.34T + 1.68e3T^{2} \) |
| 43 | \( 1 - 60.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 92.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 13.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 69.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 109.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 2.75T + 5.04e3T^{2} \) |
| 73 | \( 1 + 74.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 102. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 113.T + 9.40e3T^{2} \) |
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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
−9.699720849248142791994585194389, −8.938733865128316013291192874188, −8.049237872704457114565913866474, −7.38964773264650445087504347891, −6.34213498225875010885318137457, −5.30162232828701735673827947278, −4.61867698332118855646727105591, −3.52358821296750239129854453792, −2.60404223401611990245123851250, −1.11543141419346743369423962719,
0.41198696907216971831280711938, 1.66535959318687332481399841392, 2.23711782444029473533531076178, 3.71685707841574646948879834225, 4.81845942068836165088080856855, 6.24960005795072634852798142551, 6.54905765641855728682474266085, 7.38590987690828863600442375214, 8.305600175343928050846089841576, 8.615097720953049605514974110675
