| L(s) = 1 | + (−2.12 − 2.12i)3-s + (−10.2 + 10.2i)5-s − 32.8i·7-s + 8.99i·9-s + (18.2 − 18.2i)11-s + (−22.5 − 22.5i)13-s + 43.6·15-s + 50.1·17-s + (6.68 + 6.68i)19-s + (−69.6 + 69.6i)21-s + 186. i·23-s − 86.9i·25-s + (19.0 − 19.0i)27-s + (−118. − 118. i)29-s − 250.·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.920 + 0.920i)5-s − 1.77i·7-s + 0.333i·9-s + (0.499 − 0.499i)11-s + (−0.481 − 0.481i)13-s + 0.751·15-s + 0.715·17-s + (0.0807 + 0.0807i)19-s + (−0.723 + 0.723i)21-s + 1.69i·23-s − 0.695i·25-s + (0.136 − 0.136i)27-s + (−0.757 − 0.757i)29-s − 1.44·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2822898023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2822898023\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.12 + 2.12i)T \) |
| good | 5 | \( 1 + (10.2 - 10.2i)T - 125iT^{2} \) |
| 7 | \( 1 + 32.8iT - 343T^{2} \) |
| 11 | \( 1 + (-18.2 + 18.2i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (22.5 + 22.5i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 50.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-6.68 - 6.68i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 186. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (118. + 118. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (198. - 198. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 186. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (10.9 - 10.9i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 23.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (134. - 134. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (220. - 220. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-453. - 453. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (-184. - 184. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 18.8iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 828. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-173. - 173. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 335. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 687.T + 9.12e5T^{2} \) |
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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
−11.23164757200565676780803245007, −10.52409070159097181067543097011, −9.645510985226798317939212013903, −7.920262162468146222289300836288, −7.43956395628021480711644880267, −6.78947313169949624462167181974, −5.47925529274080961708639725010, −3.96752779954776701480040112568, −3.32664543652078290454281895667, −1.21320700559246759635267826493,
0.11227095319387264486836759432, 2.02359045644224056792881093745, 3.66043450379563259944806491994, 4.83424788926496941501972493587, 5.47789468064603997096225389329, 6.73960909223487602741990014115, 8.015468154873787798838815247392, 8.994963910018813179997096428103, 9.344371694774324882385599338411, 10.76991494611117397128171837616
