| L(s) = 1 | − 2.82i·2-s + 5.19·3-s − 8.00·4-s + 13.0·5-s − 14.6i·6-s + 71.9·7-s + 22.6i·8-s + 27·9-s − 36.8i·10-s + 76.6i·11-s − 41.5·12-s + 118. i·13-s − 203. i·14-s + 67.7·15-s + 64.0·16-s + 31.5·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.500·4-s + 0.521·5-s − 0.408i·6-s + 1.46·7-s + 0.353i·8-s + 0.333·9-s − 0.368i·10-s + 0.633i·11-s − 0.288·12-s + 0.701i·13-s − 1.03i·14-s + 0.301·15-s + 0.250·16-s + 0.109·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.140028817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.140028817\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (-729. + 3.40e3i)T \) |
| good | 5 | \( 1 - 13.0T + 625T^{2} \) |
| 7 | \( 1 - 71.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 76.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 118. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 31.5T + 8.35e4T^{2} \) |
| 19 | \( 1 - 461.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 901. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.16e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 457. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 440. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.19e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.24e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.31e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 53.4T + 7.89e6T^{2} \) |
| 61 | \( 1 + 235. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.94e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 7.82e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.26e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 519.T + 3.89e7T^{2} \) |
| 83 | \( 1 - 6.91e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.20e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.07e4iT - 8.85e7T^{2} \) |
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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
−11.00164356019501577780760225320, −9.576502494361268109492082708176, −9.383051106930569379900929423137, −7.983543910688213327964896869842, −7.38305408473134378808462409734, −5.62189779717656514467087658461, −4.70597503064167213152313029507, −3.56373316128748075121510159215, −2.04732316318599916995800025273, −1.43591466481139717065239191111,
0.950873101005006001734548621637, 2.38469371086603567804605209389, 3.91054311265961732005371681408, 5.13546647509543404092263966785, 5.88187787243087943930991556731, 7.31833270608690662222131714577, 8.040873006162424247926595571200, 8.755121921743988311229029112701, 9.768953658694491261656819929160, 10.76845362068138373830272862404
