L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.64i·7-s − 2.82·8-s − 2.32i·11-s − 18.6i·13-s + 3.74i·14-s + 4.00·16-s + 12.0·17-s − 30.0·19-s + 3.29i·22-s − 21.5·23-s + 26.4i·26-s − 5.29i·28-s − 23.6i·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.377i·7-s − 0.353·8-s − 0.211i·11-s − 1.43i·13-s + 0.267i·14-s + 0.250·16-s + 0.706·17-s − 1.58·19-s + 0.149i·22-s − 0.937·23-s + 1.01i·26-s − 0.188i·28-s − 0.814i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5598764746\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5598764746\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 11 | \( 1 + 2.32iT - 121T^{2} \) |
| 13 | \( 1 + 18.6iT - 169T^{2} \) |
| 17 | \( 1 - 12.0T + 289T^{2} \) |
| 19 | \( 1 + 30.0T + 361T^{2} \) |
| 23 | \( 1 + 21.5T + 529T^{2} \) |
| 29 | \( 1 + 23.6iT - 841T^{2} \) |
| 31 | \( 1 + 55.0T + 961T^{2} \) |
| 37 | \( 1 - 38.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 71.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 80.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 71.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 51.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 1.95T + 3.72e3T^{2} \) |
| 67 | \( 1 + 67.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 56.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 126. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 35.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 39.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 56.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 129. iT - 9.40e3T^{2} \) |
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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
−8.479598573567606575746729550501, −7.997724475841473785413183705812, −7.47201549656782014275942730782, −6.35356700648301836366332256391, −5.91320800969175535790252127798, −4.87337927256716485309266192980, −3.83222182631220300446224268508, −2.99496643088488716374437214181, −1.95295200478049039619329132319, −0.812333214028260789780106008860,
0.19476172534054218785287757565, 1.78546442889380614906303049032, 2.20730037382501535196588801065, 3.61756402226634345587532190726, 4.31246383905650296489168556020, 5.49727811537413690166471279743, 6.13301289569100948352884322216, 7.07180401349994645107814259949, 7.48623975060821184466115070887, 8.666218074834490488463220394842