| L(s) = 1 | − 2.82i·2-s − 5.19·3-s − 8.00·4-s + 48.7·5-s + 14.6i·6-s + 6.09·7-s + 22.6i·8-s + 27·9-s − 137. i·10-s − 148. i·11-s + 41.5·12-s − 20.5i·13-s − 17.2i·14-s − 253.·15-s + 64.0·16-s + 167.·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.500·4-s + 1.94·5-s + 0.408i·6-s + 0.124·7-s + 0.353i·8-s + 0.333·9-s − 1.37i·10-s − 1.22i·11-s + 0.288·12-s − 0.121i·13-s − 0.0880i·14-s − 1.12·15-s + 0.250·16-s + 0.580·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00321 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.00321 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.319985953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319985953\) |
| \(L(3)\) |
|
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 | \( 1 + (3.48e3 - 11.1i)T \) |
| good | 5 | \( 1 - 48.7T + 625T^{2} \) |
| 7 | \( 1 - 6.09T + 2.40e3T^{2} \) |
| 11 | \( 1 + 148. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 20.5iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 167.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 313.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 805. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.25e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.45e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.71e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.55e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.85e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.60e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.49e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 3.27e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.24e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 9.03e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.45e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 5.24e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 8.25e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.41e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.13e4iT - 8.85e7T^{2} \) |
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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
−10.73327920329477389209456207564, −9.719581653351486817461068111961, −9.228162689424306899855133364282, −7.933340885342933475869940130876, −6.19021813615543092727774228647, −5.86258401402323980663493079961, −4.75760914129046907663553036213, −3.11636773049040011888435728166, −1.90977535645599283486941381517, −0.825919212270408204892861509926,
1.24077912134022979885617660098, 2.46605635075175026727352914852, 4.62412482703100679201510308647, 5.25116214252413515219114592080, 6.47525931425215705142454071912, 6.70170730451201821300182808843, 8.303689742665280353673141865887, 9.328129996237451934551492319538, 10.13789559496437584281147453786, 10.62324379951902823863349151445
